CUT POINTS IN GROUPS (FIRST DRAFT)

KATSUYA EDA

1. Introduction

A new construction of spaces from groups was introduced in [1] and it was shown that for the fundamental groups of one dimensional wild Peano continua and countable direct products of one dimensional wild Peano continua the construction produces the original spaces. This implies that some topological notions have their group theoretic correspondence. In this paper we concern cut points in spaces and as a consequece we prove that in addition to the spaces mentioned precedingly, spaces obtained by iterated use of one point unions and countable direct products can be recovered by the construction from their fundamental groups. To do this we generalize the notion of free products to an infinitary variant. On this line we generalize the free σ-product in [3] so that we can treat with more general one point unions of spaces. The construction in [1] is meaningful for groups in a certain class and in the remaining part of this section we review the notion of n-slender groups [3] and the class of groups treated in [1]. The free σ-product × ×_{n<ω}ℤ_{n} for copies ℤ_{n} of the integer group is isomorphic to the fundamental group of the Hawaiian earring [3, Appendix]. The generator of ℤ_{n} is denoted by δ_{n} and corresponds to a winding e_{n} to the n-th circle of the Hawaiian earring. By “n < ω” we mean n is a non-negative integer.

Definition 1.1. [3, Definition 3.1] A group S is n-slender, if for each homomorphism h : × ×_{n<ω}ℤ_{n} → S the set {n < ω : h(δ_{n}) ≠ e} is finite.

The class S of groups consists of all the groups G such that for any non-trivial element g ∈ G there exist an n-slender group S and a homomorphism h : G → S with h(g) ≠ e.

Since a finite direct product of n-slender groups are again n-slender [3, Theorem 3.6], we have,

Proposition 1.2. The following are equivalent for a group G: (1) G belongs to S;

(2) G is a subgroup of a direct product of n-slender groups;

(3) G is a subgroup of an inverse limit of n-slender groups.

Also the following are equivalent for a group G:

(1’) G belongs to S^{σ};

(2’) G is a subgroup of a countable direct product of n-slender groups; (3’) G is a subgroup of an inverse limit of a countable sequence of

n-slender groups.

Proposition 1.3. Let G ∈ S and h_{0} and h_{1} be homomorphisms from × ×_{n<ω}ℤ_{n} to G. If h_{0}(δ_{n}) ≠ e for any n < ω, then the cardinality of Im(h_{0}) is 2^{ℵ_{0}}. If h_{0}(δ_{n}) = h_{1}(δ_{n}) for every n < ω, then h_{0} = h_{1}.

Proof. We choose homomorphisms g_{n} from G to n-slender groups S_{n} and a strictly increasing sequence i_{n} < i_{n+1} inductively as follows: Let i_{0} = 0. For a defined i_{n} let g_{n} be a homomorphism from G to nslender groups S_{n} such that g_{n}(h(δ_{i_{n}})) ≠ e, which can be done by the assumption G ∈ S. Next by the n-slenderness of S_{n} we have i_{n+1} such that g_{n}(x) = e for x ∈ × ×_{i≥i_{n}}ℤ_{i}. Then it is easy to see that h is injective on a set {δ_{i_{0}}^{ε_{0}} ⋯ δ_{i_{n}}^{ε_{n}} ⋯: ε_{n} = ±1} of the cardinality 2^{ℵ_{0}}.

To show h_{0} = h_{1} by contradiction, suppose that there exists x such that h_{0}(x) ≠ h_{1}(x). Then we have a homomorphism g from G to an n-slender group S with g(h_{0}(x)^{−}h_{1}(x)) ≠ e. By the n-slenderness of S we have a canonical projection p_{m} : × ×_{n<ω}ℤ_{n} → ∗_{n<m}ℤ_{n} and homomorphisms g_{0}, g_{1} : ∗_{n<m}ℤ_{n} → S such that g · h_{0} = g_{0} · p_{n} and g · h_{1} = g_{1} · p_{n}. Since g · h_{0} | ∗_{n<m} ℤ_{n} = g · h_{1} | ∗_{n<m} ℤ_{n}, g_{0} = g_{1} and hence g(h_{0}(x)) = g_{0} · p_{n}(x) = g_{1} · p_{n}(x) = g(h_{1}(x)), which is a contradiction. □

Let H_{G} be the set of all subgroups of a group G which are homomorphic images of × ×_{ω}ℤ. A subfamily x of H_{G} is compatible, if for every finite subset F of x there exists H ∈ H_{G} such that ⋃ F ⊆ H. Let X_{G} be the set of all maximal compatible subfamilies of H_{G} which contain uncountable subgroups of G. For subgroups H, H′ of G, H ⪯ H′ if there exists a finite subset F of G such that H ≤ ⟨H′ ∪ F ⟩ holds. We note that for x ∈ X_{G}, H ∈ x and H′ ⪯ H imply H′ ∈ x for H, H′ ∈ H(G).

Topology on the set X_{G} is defined as follows: For Y ⊆ X_{G}, Y is closed if each point x with the following property belongs to Y; there exists a sequence (x_{n} : n < ω) of elements of Y satisfying the following condition (*).

(*) For given uncountable H_{n} ∈ x_{n} (n < ω) there exist H′_{n} ∈ x_{n} satisfying the following:

(1) H_{n} ⪯ H′_{n};

(2) for arbitrary a_{n} ∈ H′_{n} (n < ω) there exists h : × ×_{n<ω}ℤ_{n} → G such that h(δ_{n}) = a_{n} for every n < ω and Im(h) ∈ x.

In general X_{G} may not satisfy any separation axiom, but we write lim_{n→∞} x_{n} = x when (*) holds. We remark that lim_{n→∞} x_{n} = x implies lim_{k→∞} x_{n_{k}} = x for every subsequence (x_{n_{k}} : k < ω) of (x_{n} : n < ω). We refer the reader to [1] for further information for the topology of X_{G}.

2. Infinitary words and infinitary variants of free products

First we define words in an infinitary sense [3, 5]. For a set D let D^{−} be the set of formal inverses d^{−} for d ∈ D, i.e. {d^{−} : d ∈ D}∩D = ∅ and d^{−} ≠ d′^{−} for d ≠ d′. We identify d^{−−} with d. Let W(D) be the set of words W on D, i.e. \overline{W} is a linearly ordered set, W : \overline{W} → D∪\{d^{−} : d ∈ D\} and \{α ∈ \overline{W} : W (α) = u\} is finite for each u ∈ D ∪ \{d^{−} : d ∈ D\}. The class of all words on D is denoted by W(D) and the subclass of W(D) consisting of all countable words is denoted by W^{σ}(D). We remark W^{σ}(D) = W(D), if D is countable. Let W^{−} be the inverse word of W, i.e. \overline{W^{−}} is the reversedly ordered set of \overline{W} and W^{−}(α) = W (α)^{−} for α ∈ \overline{W}. When a subset of D forms a group, we also identify d^{−} with the inverse d^{−1} of d. Hence we express the inverse of an element g of a group by g^{−}.

For groups G_{i} (i ∈ I) with G_{i} ∩ G_{j} = {e} (i ≠ j), W^{σ}(G_{i} : i ∈ I) is the class of words W ∈ W^{σ}(⋃_{i∈I} G_{i}) such that \{α ∈ \overline{W} : W (α) ∈ G_{i}\} is finite for each i. We define a new subclass of W^{σ}(⋃_{i∈I} G_{i}): W_{adm}^{σ}(G_{i} : i ∈ I) consists of W such that for each homomorphism h from ∗_{i∈F} G_{i} to an n-slender group S the set \{α ∈ \overline{W} : W (α) ∈ ⋃_{i∈F} G_{i}, h(W (α)) ≠ e\} is finite. (We remark that W_{adm}^{σ}(G_{i} : i ∈ I) is much larger than W^{σ}(G_{i} : i ∈ I) in general.)

The identification of two words is given by the following. For U, V ∈ W(D), U ≡ V if there exists an order-isomorphism φ : \overline{U} → \overline{V} such that U(α) = V (α) for α ∈ \overline{U}. A subset S of a linearly ordered set is a segment, if s_{0} ≤ x ≤ s_{1} implies x ∈ S for each s_{0}, s_{1} ∈ S. A word Y is a subword of U, if there exist words X and Z such that U ≡ XY Z, that is, \overline{U} is the sum of segments \overline{X}, \overline{Y} , \overline{Z}, x < y < z for any x ∈ \overline{X}, y ∈ \overline{Y} , z ∈ \overline{Z} and X, Y, Z are restrictions of U.

Definition 2.1. At most countable subset C of a group G is admissible, if there exists a homomorphism h : × ×_{n<ω}ℤ_{n} → G such that C ⊆ {h(δ_{n}) : n < ω}.

A subgroup H of a group G is admissibly closed, if for each admissible subset C of G with C ⊆ H there exists a homomorphism h : × ×_{n<ω}ℤ_{n} → H such that C ⊆ {h(δ_{n}) : n < ω}.

We recall that a sequence (x_{n} : n < ω) in × ×_{n<ω}ℤ_{n} is proper if the numbers of appearances of δ_{m}^{±1} in the reduced words for x_{n}’s are bounded for each m [5, Definition 2.1].

A family c of admissible subsets is compatible, if ⋃ F is also admissible for each finite subset F of c and c is closed under taking subsets, i.e. C ∈ c and D ⊆ C imply D ∈ c. A compatible family of admissible subsets c is properly closed, if the following hold: if (x_{n} : n < ω) is a proper sequence in × ×_{n<ω}ℤ_{n} and {h(δ_{n}) : n < ω} ∈ c for h : × ×_{n<ω}ℤ_{n} → G, then {h(x_{n}) : n < ω} ∈ c. A compatible family c is maximal, if c is maximal with respect to the set theoretic inclusion among compatible families.

Proposition 2.2. Every finite subset F of a group G is admissible.

Proof. Since × ×_{i<ω}ℤ_{i} ≃ ∗_{i<n}ℤ_{i} ∗ × ×_{i≥n}ℤ_{i} for each n, we have a homomorphism h : × ×_{n<ω}ℤ_{n} → G such that F = {h(δ_{i}) : i < n} and h(× ×_{i≥n}ℤ_{i}) = {e}. □

According to this, the set of all finite subsets of G is a properly closed family of admissible subsets. Let C_{G} be the set of all properly closed compatible families c of admissible subsets such that c contains all finite subsets of G. Hence, the set of all finite subsets of G is the minimal element of C_{G}.

Even for a group G ∈ S a relationship between elements of X_{G} and maximal compatible elements of C_{G} is unclear so far, but in Section 6 we shall see a clear correspondence between them for a group in a more restricted class. The next proposition is a restatement of the definition of the n-slenderness.

Proposition 2.3. If G is n-slender, then every admissible subset of G is finite.

Lemma 2.4. Let {x_{n} : n < ω} be an admissible set in a group G and {n < ω : x_{n} = x} is finite for each x ≠ e. Then there exists a homomorphism h : × ×_{n<ω}ℤ_{n} → G such that h(δ_{n}) = x_{n}.

Proof. Let g : × ×_{n<ω}ℤ_{n} → G be a homomorphism such that {x_{n} : n < ω} ⊆ {g(δ_{n}) : n < ω}. Choose i_{x} < ω so that g(δ_{i_{x}}) = x for x ∈ {x_{n} : n < ω} and let y_{n} = δ_{i_{xn}}. Then (y_{n} : n < ω) is a proper sequence in × ×_{n<ω}ℤ_{n} and we have an endomorphism φ on × ×_{n<ω}ℤ_{n} such that φ(δ_{n}) = y_{n} by [3, Proposition 1.9]. Now g · φ(δ_{n}) = x_{n}. □

Lemma 2.5. Let c ∈ C_{G} and {x_{n} : n < ω}, {y_{n} : n < ω} ∈ c. Then {x_{n}^{ε_{n}} : n < ω} ∈ c for any ε_{n} = ±1. If {n < ω : x_{n} = x} and {n < ω : y_{n} = x} are finite for x ≠ e, then {x_{n}y_{n} : n < ω} ∈ c.

Proof. Since (δ_{n}^{ε_{n}} : n < ω) is a proper sequence, the first statement is clear by [3, Proposition 1.9]. We prove the second. Since c is a compatible family and {x_{n}, y_{n} : n < ω} is infinite, we have {x_{n}, y_{n} : n < ω} ∈ c by the compatibility of c. By Lemma 2.4 there exists a homomorphism h : × ×_{n<ω}ℤ_{n} → G such that h(δ_{2n}) = x_{n}, h(δ_{2n+1}) = y_{n}. Since (δ_{2n}δ_{2n+1} : n < ω) is a proper sequence, we have the conclusion.

□

Lemma 2.6. If a compatible family c is maximal, then c ∈ C_{G}.

Proof. Let h : × ×_{n<ω}ℤ_{n} → G, {h(δ_{n}) : n < ω} ∈ c and (x_{n} : n < ω) be a proper sequence in × ×_{n<ω}ℤ_{n}. Let y_{2n} = δ_{n} and y_{2n+1} = x_{n}. Since (y_{n} : n < ω) is a proper sequence, there exists an endomorphism g on × ×_{n<ω}ℤ_{n} such that g(δ_{n}) = y_{n} for each n by [3, Proposition 1.9]. Then h·g witnesses the compatibility of {h(x_{n}) : n < ω} and {h(δ_{n}) : n < ω}. Since c is maximal, it suffices to show that {h(x_{n}) : n < ω} ∪ ⋃ F is admissible for each F ⋐ c. This easily follows from the above fact. □

Lemma 2.7. Let G_{i} ∈ S, C_{i} be an admissible subset of G_{i} for each i, U ∈ W(⋃_{i∈I} C_{i}) and S be an n-slender group. For a homomorphism h : ∗_{i∈F} G_{i} → S for a finite subset F of I, the set \{α ∈ \overline{U} : h(U(α)) ≠ e\} is finite.

Proof. There exists a homomorphism g_{i} : × ×_{n<ω}ℤ_{n} → G_{i} such that {g_{i}(δ_{n}) : n < ω} = C_{i} for each i ∈ F. Since {n < ω : h · g_{i}(δ_{n}) ≠ e} is finite, \{α ∈ \overline{U} : U(α) ∈ G_{i}, h(U(α)) ≠ e\} is finite for each i ∈ F and we have the conclusion. □

By this lemma \{α ∈ \overline{U} : h(U(α)) ≠ e\} is finite. We define U_{F}^{h} by: \overline{U_{F}^{h}} = \{α ∈ \overline{U} : h(U_{α}) ≠ e\} and \overline{U_{F}^{h}}(α) = U(α). Since U_{F}^{h} is finite, U_{F}^{h} presents an element of ∗_{i∈F} G_{i} and h(U_{F}^{h}) is defined. We define h(U) ∈ S to be h(U_{F}^{h}).

For homomorphisms h_{i} : G_{i} → S_{i} we denote the canonical homomorphism h : ∗_{i∈F} G_{i} → ∗_{i∈F} S_{i} by h = (h_{i} : i ∈ F), i.e. h ↾ G_{i} = h_{i} for each i.

Lemma 2.8. Let G ∈ S, C be an admissible subset of G and U ∈ W(C). Then there exists a unique u ∈ G such that h(u) = h(U) for every homomorphism h from G to every n-slender group S. Moreover such u s form a subgroup of G.

Proof. Let h_{0} : × ×_{n<ω}ℤ_{n} → G be a homomorphism such that {h_{0}(δ_{n}) : n < ω\} = C and W ∈ W^{σ}(ℤ_{n} : n < ω) such that \overline{W} = \overline{U} and h_{0}(W (α)) = U(α) for α ∈ \overline{W}. Since W presents an element of × ×_{n<ω}ℤ_{n}, we have u = h_{0}(W) ∈ G. Let \{α ∈ \overline{U} : h(U(α)) ≠ e\} = \{α_{0}, ⋯, α_{n}\} so that α_{0} < ⋯ < α_{n} and U ≡ U_{0}U(α_{0})U_{1}U(α_{1}) ⋯ U(α_{n})U_{n+1}. According to this decomposition we have W ≡ W_{0}W (α_{0})W_{1}W (α_{1}) ⋯ W (α_{n})W_{n+1}. Since S is n-slender and h · h_{0}(W_{i}(α)) = e for α ∈ \overline{W_{i}}, h · h_{0}(W_{i}) = e for each i. Now u = h_{0}(W_{0})U(α_{0})h_{0}(W_{1})U(α_{1}) ⋯ U(α_{n})h_{0}(W_{n+1}) and hence h(u) = h(U(α_{0}))h(U(α_{1})) ⋯ h(U(α_{n})) = h(U). The uniqueness of u follows from G ∈ S. Since W (C) is closed under concatenation and taking the inverse words, the remaining statement is clear. □

For an admissible subset C of G ∈ S we let H(C) to be the subgroup of G according to this Lemma 2.8.

Lemma 2.9. Let G ∈ S. A subgroup H of G is admissibly closed if and only if H(C) ⊆ H for each admissible subset C of G with C ⊆ H.

Proof. If H is admissibly closed, there exists a homomorphism h_{0} : × ×_{n<ω}ℤ_{n} → H such that C = {h_{0}(δ_{n}) : n < ω}. We let W be a word defined in the proof of Lemma 2.8. Then u = h_{0}(W) and hence u ∈ H, that is H(C) ⊆ H.

To show the remaining direction of the equivalence, let h : × ×_{n<ω}ℤ_{n} → G be a homomorphism such that C = {h_{0}(δ_{n}) : n < ω}. For each W ∈ W^{σ}(ℤ_{n} : n < ω), define U by: \overline{U} = \overline{W} and U(α) = h(W (α)). Then U belongs to W(C) and h(W) is the corresponding element to U in Lemma 2.8, which implies h(W) ∈ H, that is, h maps into H. □

We define classes of words together with an equivalence relation.

Definition 2.10. We assume G_{i} ∩ G_{j} = {e} for i ≠ j. Let W_{adm}^{σ}(G_{i} : i ∈ I) be the subclass of W^{σ}(⋃_{i∈I} G_{i}) consisting of words W such that \{α ∈ \overline{W} : W (α) ∈ ⋃_{i∈F} G_{i}, g(W (α)) ≠ e\} is finite for each nslender group S and each homomorphism h : ∗_{i∈F} G_{i} → S (F ⋐ I). For c_{i} ∈ C_{G_{i}} we let ⋓(c_{i} : i ∈ I) to be the set of all sets C such that C ⊂ ⋃{C_{i} : i ∈ I} for C_{i} ∈ c_{i} and ⋓^{F}(c_{i} : i ∈ I) to be the set all sets C such that C ⊂ ⋃{C_{i} : i ∈ F } for C_{i} ∈ c_{i} and F ⋐ I. Let W^{σ}(G_{i}, c_{i} : i ∈ I) be the class of words W ∈ W^{σ}(C) for some C ∈ ⋓(c_{i} : i ∈ I) and W^{fin}(G_{i}, c_{i} : i ∈ I) its subclass consisting of words W ∈ W^{σ}(C) for some C ∈ ⋓^{F}(c_{i} : i ∈ I).

Proposition 2.11. Let c_{i} ∈ C_{G_{i}} for each i ∈ I. Then W^{σ}(G_{i}, c_{i} : i ∈ I) is a subclass of W_{adm}^{σ}(G_{i} : i ∈ I).

Proof. Let W ∈ W^{σ}(G_{i}, c_{i} : i ∈ I) and h : ∗_{i∈F} G_{i} → S be a homomorphism for an n-slender group S and F ⋐ I. There exist C_{i} ∈ c_{i} (i ∈ I) such that W ∈ W^{σ}(⋃_{i∈I} C_{i}). By the definition of c_{i}’s the set {x ∈ C_{i} : h(x) ≠ e} is finite for each i ∈ F and consequently the set \{α ∈ \overline{W} : W (α) ∈ ⋃_{i∈F} G_{i}, g(W (α)) ≠ e\} is finite. □

Definition 2.12. Let G_{i} be a group in S. Words U, V ∈ W_{adm}^{σ}(G_{i} : i ∈ I) are weakly equivalent, if h(U) = h(V) holds for each n-slender group S, homomorphism h : ∗_{i∈F} G_{i} → S (F ⋐ I). In this case, we write U ∼_{w} V.

A word U ∈ W_{adm}^{σ}(G_{i} : i ∈ I) is reduced with respect to ∼_{w} (∼_{w}- reduced for short), if

(1) for a subword Y of U, Y ∼_{w} ⟨⟩ implies that Y is the empty word;

(2) elements of each G_{i} appear isolated in U, i.e. every subword of U in some W(G_{i}) consists of a single letter.

U is quasi-∼_{w}-reduced, if the property (1) is satisfied.

Lemma 2.13. Let G_{i} ∈ S, C_{i} be an admissible subset of G_{i} and U ∈ W^{σ}(⋃_{i∈I} C_{i}). If H is an admissibly closed subgroup of G_{i_{0}} containing C_{i_{0}} and U ∼_{w} u ∈ G_{i_{0}}, then u belongs to H.

Proof. Define U_{0} ∈ W(C_{i_{0}}) as: \overline{U_{0}} = \{α ∈ \overline{U} : U(α) ∈ G_{i_{0}}\} and U_{0}(α) = U(α). For each homomorphism h : G_{i_{0}} → S to an n-slender group S, we have h(U_{0}) = h(U) = h(u). Hence u belongs to H by Lemma 2.9. □

Lemma 2.14. For a ∼_{w}-reduced word U ∈ W_{adm}^{σ}(G_{i} : i ∈ I), let g_{i} : G_{i} → S_{i} be homomorphisms to n-slender groups S_{i} for i ∈ F_{0} and α_{0} < ⋯ < α_{n} be such that \{α_{0}, ⋯, α_{n}\} = \{α ∈ \overline{U} : g(U(α)) ≠ e\}, where g = (g_{i} : i ∈ F_{0}). Then there exist subwords X_{k} (0 ≤ k ≤ n+1) and Y_{k} (0 ≤ k ≤ n) and homomorphisms h_{i} : G_{i} → T_{i} to n-slender groups T_{i} for i ∈ F ⋐ I satisfying:

• U ≡ X_{0}Y_{0}X_{1} ⋯ X_{n}Y_{n}X_{n+1};

• α_{k} ∈ \overline{Y_{k}} and F_{0} ⊆ F;
• g((Y_{k})_{F}^{g}) = g(U(α_{k})) and g((X_{k})_{F}^{g}) = e;

• h_{i(k)}(U(α_{k})) ≠ e and e ≠ h((Y_{k})_{F}^{h}) = u_{k} ∈ T_{i(k)} for i(k) ∈ F

such that U(α_{k}) ∈ G_{i(k)};

• for the reduced word W_{k} for h((X_{k})_{F}^{h}), W_{0}u_{0}W_{1} ⋯ W_{n}u_{n}W_{n+1}

is a reduced word in W(T_{i} : i ∈ F).

Particularly if G_{i} ≤ Π_{m<ω}S_{im} for n-slender groups S_{im}, S_{i} = S_{i0} and g_{i} is a restriction of the projection p_{i0} from Π_{m<ω}S_{im} to S_{i0} for i ∈ F_{0}, then h_{i} can be taken as p_{im_{0}} : Π_{m<ω}S_{im} → Π_{m≤m_{0}}S_{im} for i ∈ F.

Proof. Let subwords X′_{k} of U be so that U ≡ X′_{0}U(α_{0})X′_{1} ⋯ X′_{n}U(α_{n})X′_{n+1}. Since U is ∼_{w}-reduced, X′_{k} is non-empty when U(α_{k}) and U(α_{k+1}) belong to the same G_{i}. Then we have finitely many n-slender groups S_{i} (i ∈ F) and homomorphisms f_{i} : G_{i} → S′_{i} so that F_{0} ⊆ F, i_{0} ∈ F, and f((X′_{k})_{F}^{f}) ∉ S′_{i(k)} when U(α_{k}) and U(α_{k+1}) belong to the same G_{i} where f = (f_{i} : i ∈ F). Let T_{i} = S_{i} × S′_{i} for i ∈ F where S_{i} = {e} for i ∈ F \ F_{0} and h_{i} : G_{i} → T_{i} be the homomorphism defined by h_{i}(x) = (g_{i}(x), f_{i}(x)). Expanding U(α_{k})’s and shrinking X′_{k}’s we have the desired subwords Y_{k} and X_{k} of U.

Under the additional condition T_{i} may be taken as a finite product Π_{m≤k}S_{im} and h_{i} may be taken as a restriction of the projection. □

Lemma 2.15. Let U and V be ∼_{w}-reduced words in W_{adm}^{σ}(G_{i} : i ∈ I)such that U ∼_{w} V. Then U ≡ V holds.

Proof. Let g_{i} : G_{i} → S_{i} be homomorphisms to n-slender groups S_{i} for i ∈ F_{0} and α_{0} < ⋯ < α_{m} and β_{0} < ⋯ < β_{m′} be such that \{α_{0}, ⋯, α_{m}\} = \{α ∈ \overline{U} : g(U(α)) ≠ e\} and \{β_{0}, ⋯, β_{m′}\} = \{β ∈ \overline{U} : g(U(β)) ≠ e\}, where g = (g_{i} : i ∈ F_{0}). Applying Lemma 2.14 and looking at the reduced word of h(U_{F}^{h}) = h(V_{F}^{h}), we can see that m = m′ and U(α_{k}) and V (β_{k}) belongs to the same G_{i}. To show U(α_{k}) = V (β_{k}) by contradiction, suppose U(α_{k}) ≠ V (β_{k}). We have U(α_{k}), V (β_{k}) ∈ G_{i_{0}} and a homomorphism g′ : G_{i_{0}} → S to an n-slender group S such that g′(U(α_{k})) ≠ g′(V (β_{k})). Using a homomorphism from G_{i_{0}} to S_{i} × S instead of g_{i_{0}}, we define \overline{g} : ∗_{i∈F} G_{i} → ∗_{i∈F,i≠i_{0}}S_{i} ∗ (S_{i_{0}} × S). Applying Lemma 2.14 to U, V and \overline{g}, we have a homomorphism \overline{h} and α′_{0} < ⋯ < α′_{n} and β′_{0} < ⋯ < β′_{n} so that \{α′_{0}, ⋯, α′_{n}\} = \{α ∈ \overline{U} : \overline{g}(U(α)) ≠ e\} and \{β′_{0}, ⋯, β′_{n}\} = \{β ∈ \overline{U} : \overline{g}(U(β)) ≠ e\}. (By the above argument the lengths of the two sequences are same.) Then α_{0} < ⋯ < α_{m} and β_{0} < ⋯ < β_{m} are subsequences of α′_{0} < ⋯ < α′_{n} and β′_{0} < ⋯ < β′_{n} respectively. According to the values of g we can see α_{k} = α′_{i} if and only if β_{k} = β′_{i}, which contradicts \overline{g}(U(α_{k})) ≠ \overline{g}(V (β_{k})). We define φ : \overline{U} → \overline{V} by: φ(α_{k}) = β_{k}.

Now it suffices to show that φ is well-defined. To see this suppose that we are given finitely many homomorphisms f_{i} : G_{i} → S_{i} to nslender groups S_{i} for i ∈ F_{0} and g_{i} : G_{i} → T_{i} to n-slender groups T_{i} for i ∈ F_{1}. Then we define h_{i} : G_{i} → S_{i} × T_{i} by: h_{i}(x) = (g_{i}(x), h_{i}(x)) for i ∈ F_{0} ∪ F_{1}, as in the proof of Lemma 2.14. Then apply Lemma 2.14 and we can see that f_{i}’s and g_{i}’s give the same value for φ. □

Lemma 2.16. Let W ∈ W_{adm}^{σ}(G_{i} : i ∈ I) and C ∈ c_{i_{0}} ∈ C_{G_{i_{0}}} and let i_{0} ∈ I. If U_{m}’s (m < ω) are subwords of W such that U_{m} ∈ W(C) for and U_{m} = u_{m}, then {u_{m} : m < ω} ∈ c_{i_{0}}.

Proof. Let C = {h(δ_{n}) : n < ω}) ∈ c_{i_{0}}. Since it is suffices to treat with the case that C is infinite, we may assume that h(δ_{m}) ≠ h(δ_{n}) for m ≠ n.

We define \overline{W_{m}} = \overline{U_{m}} and W_{m}(α) = δ_{n} if U_{m}(α) = h(δ_{n}) (n < ω). By Lemma 2.7 W_{m} ∈ W(ℤ_{n} : n < ω) and also l_{n}(W_{m}) = 0 for almost all m, where l_{n}(W) is the number of appearances of elements of ℤ_{n} in W. It suffices to show h(W_{m}) = u_{m}. Let g : G → S be a homomorphism to an n-slender group S. Let α_{0} < ⋯ < α_{n} be the enumeration of \{α ∈ \overline{U_{m}} : g(U_{m}) ≠ e\}. By the n-slenderness of S, g · h(W_{m}) = g · h(W_{m}(β_{0}) ⋯ W_{m}(β_{k})) for some β_{0} < ⋯ < β_{k}. Hence g · h(W_{m}) = g(U_{m}(α_{0}) ⋯ U_{m}(α_{n})) = g(U_{m}). Therefore h(W_{m}) = u_{m} for each m < ω. Since c_{i_{0}} is properly closed, {u_{m} : m < ω} ∈ c_{i_{0}}. □

Theorem 2.17. Let G_{i}’s be groups in S. For a word W ∈ W_{adm}^{σ}(G_{i} : i ∈ I), there exists a unique ∼_{w}-reduced word V ∈ W_{adm}^{σ}(G_{i} : i ∈ I) such that V ∼_{w} W.

Proof. We define words W_{μ} for ordinals μ by induction. Let W_{0} be W. When W_{μ} has been constructed and there exists a non-empty word X such that UXV ≡ W_{μ} and X ∼_{w} ⟨⟩, let \overline{W_{μ+1}} = \{α ∈ \overline{W_{μ}} : i(α) ∈ \overline{U} or i(α) ∈ \overline{V} \} ⊂ \overline{W} and W_{μ+1}(α) = W (α) for α ∈ \overline{W_{μ+1}}, where the ordering is the restriction of that of \overline{W} and i : \overline{W_{μ}} → \overline{UXV} is the order isomorphism. Otherwise, the procedure is completed. For a limit ordinal μ, let \overline{W_{μ}} = ⋂_{ν<μ} \overline{W_{ν}} and W_{μ}(α) = W (α) for α ∈ \overline{W}_{μ}. This procedure must stop at some countable ordinal, because the cardinality of \overline{W} is countable. Let W_{∞} be the obtained word. By induction we can see that W_{μ} ∼_{w} W. Since it is clear for a successor μ, we treat with the limit case. Let S be an n-slender group, h : ∗_{i∈I}G_{i} → S a homomorphism and F ⋐ I. Since \overline{W_{F}^{h}} = \{α ∈ \overline{W} : W (α) ∈ ⋃_{i∈F} G_{i}, h(W (α)) ≠ e} is finite and all W_{ν}’s are restrictions of W, W_{μF}^{h} is exactly the same word as W_{νF}^{h} for some ν < μ. Consequently we have h(W_{μF}^{h}) = h(W_{νF}^{h}) = h(W_{F}^{h}). When this construction stops, we have a word W_{∞} which satisfies the property (1) of the _{s}im-reducedness. Then we express W_{∞} as W_{∞} ≡ ⋯ U_{α} ⋯ U_{β} ⋯ for a linearly ordered set L so that

• each U_{α} ∈ W(G_{i}) for some i;

• each subword V of W such that V ∈ W(G_{i}) for some i is a

subword of one of U_{α}’s. We have u_{α} ∈ G_{i} which is corresponding to U_{α} and uniquely determined by Lemma 2.8. Let \overline{V} = L and V (α) = u_{α} for α ∈ L. It is easy to see that V satisfies the properties (1) and (2) of the the _{s}im-reducedness. The uniqueness follows from Lemma 2.15. □

The next corollary is a corollary to the proof of Theorem 2.17.

Corollary 2.18. Let G_{i}’s be groups in S and fix c_{i} ∈ C_{G_{i}} for each i ∈ I. For a word W ∈ W(G_{i}, c_{i} : i ∈ I), there exist a quasi ∼_{w}-reduced word V ∈ W(G_{i}, c_{i} : i ∈ I) and subwords X_{α}’s of W such that

• \overline{W} is the disjoint union of \overline{X_{α}}’s and \overline{V},
• X_{α} ∼_{w} ⟨⟩,
• V (α) = W (α) for α ∈ \overline{V}.

Consequently W ∼_{w} V holds.

Proof. We refine the proof of Theorem 2.17 as follows. We choose X_{μ} so that UX_{μ}Y ≡ W_{μ} and X_{μ} ∼_{w} ⟨⟩ and X_{μ} is maximal among those satisfying these two properties. It suffices to show that all X_{μ}’s are subwords of W. To see this, suppose the negation and let μ be the minimal ordinal such that X_{μ} is not a subword. Let \overline{Y} = \{α ∈ \overline{W} : γ ≤ α ≤ δ for some γ, δ ∈ \overline{X_{μ}}\} and Y (α) = W (α). Then Y is a subword of W and Y ∼_{w} ⟨⟩. This contradicts the construction of X_{ν} for the minimal ν such that \overline{X_{ν}} ⊆ \overline{Y}. Then the resulting W_{∞} is the desired V. □

The next corollary is a variant of [3, Corollaries 1.5, 1.6, 1.7], which was important to investigate group theoretic properties using word theoretic arguments.

Corollary 2.19. Let G_{i}’s be groups in S and words U, V ∈ W_{adm}^{σ}(G_{i} : i ∈ I) be ∼_{w}-reduced.

(1) If UV = e, then V ≡ U^{−}.

(2) If U is non-empty, U ≠ U^{−}.

(3) There exist ∼_{w}-reduced words X, Y, Z such that U ≡ XY and V ≡ Y^{−}Z and XZ is quasi ∼_{w}-reduced.

Proof. (1) Since UV = e implies U = V^{−} and V^{−} is also ∼_{w}-reduced, the first proposition is clear from Theorem 2.17.

(2) To show the second proposition by contradiction, suppose that U = U^{−}. Then U ≡ U^{−} by Theorem 2.17. We have a subword X of U such that XX^{−} ≡ U or XgX^{−} ≡ U for some letter g. Since U is ∼_{w}-reduced, XX^{−} ≡ U does not hold and we have XgX^{−} ≡ U. But this implies g = g^{−} which contradicts that each G_{i} is torsion-free.

(3) Take X to be a maximal subword so that U ≡ XY and V ≡ Y^{−}Z. To show that XZ is ∼_{w}-reduced by contradiction, suppose that W is a non-empty subword of XZ and W ∼_{w} ⟨⟩. Since U and V are ∼_{w}-reduced, W is decomposed to non-empty subwords W_{0} and W_{1} so that W ≡ W_{0}W_{1}, X ≡ X_{0}W_{0} and Z ≡ W_{1}Z_{0}. Since W_{0} and W_{1} are ∼_{w}-reduced, W_{1} ≡ W_{0}^{−} by (1), which contradicts the maximality of X. □

For words U, V, X and letters x, y, z with xy = z, we call the transformation from UXX^{−}V to UV as a cancellation and the transformation from UxyV to UzV as a multiplication. Though these transformations are not sufficient to get a ∼_{w}-reduced word from a word in general, we can obtain the ∼_{w}-reduced word of a word consisting of finitely many ∼_{w}-reduced words by the use of finitely many times of these transformations by Corollary 2.19(3). The proof of the next corollary is the same as that of [5, Lemma 2.4] and we omit it.

Corollary 2.20. Let G_{i}’s be groups in S. For a non-trivial x ∈ × ×_{adm}^{σ}(G_{i} : i ∈ I) there exist unique ∼_{w}-reduced words W and V such that

(1) x = W^{−}V W and W^{−}V W is quasi ∼_{w}-reduced;

(2) V W is ∼_{w}-reduced;

(3) if V is not a single letter, then V V is ∼_{w}-reduced.

Definition 2.21. A subword X of a word W is stable in W, if the following hold:

• W ≡ Y XZ;

• the ∼_{w}-reduced word of W is of the form Y′XZ′ where Y′ and Z′ are the ∼_{w}-reduced word of Y and Z respectively;

• there is no subword V of W for which V = e and V intersects with X.

We remark about the last property in this definition. Let X ≡ aa and W ≡ Xaa^{−}a^{−}a. In this case X satisfies the first and second property, but does not satisfy the last, since aaa^{−}a^{−} = e. In other words, a subword of W is stable, if it remains in the ∼_{w}-reduced word of W in any reduction. When no confusion would occur, we simply say “stable” instead of “stable in W ”.

Definition 2.22. We call a canonical presentation of x for W^{−}V W in Corollary 2.20. A non-empty word V is cyclically ∼_{w}-reduced, if V is a single letter or V V is ∼_{w}-reduced. A word W is cyclically equivalent to a word V, if W ≡ XY and V ≡ Y X for some words X, Y.

Lemma 2.23. Let x = W_{x}^{−}V_{x}W_{x} and y = W_{y}^{−}V_{y}W_{y} be canonical presentations. Then V_{x} is cyclically equivalent to V_{y} if and only if x is conjugate to y. Consequently, when x is conjugate to y, V_{x} is a single letter if and only if V_{y} is a single letter.

Proof. If V_{x} is cyclically equivalent to V_{y}, we have words X, Y such that V_{x} ≡ XY and V_{y} ≡ Y X. Since V_{x} = Y^{−}Y XY,

x = W_{x}^{−}Y^{−}Y XY W_{x} = W_{x}^{−}Y^{−}Y XY W_{x}
= W_{x}^{−}Y^{−}W_{y}W_{y}^{−}Y XW_{y}W_{v}^{−}Y W_{x} = (W_{y}W_{v}^{−}Y W_{x})^{−}y(W_{y}W_{v}^{−}Y W_{x}).

On the other hand, if x is conjugate to y, we have a word U_{0} such that x = U_{0}^{−}W_{y}^{−}V_{y}W_{y}U_{0}. Let U_{1} be the ∼_{w}-reduced word of W_{y}U_{0}.

If V_{y}U_{1} is ∼_{w}-reduced and U_{1}^{−}V_{y} is quasi ∼_{w}-reduced, then W_{x} ≡ U_{1} and V_{x} ≡ V_{y} by the fact that V_{x} and V_{y} are cyclically ∼_{w}-reduced. Otherwise. If V_{y}U_{1} is not ∼_{w}- reduced, we have X_{0}, Y_{0}, Z_{0} such that Y_{0}X_{0} ≡ V_{y}, X_{0}^{−}Z_{0} ≡ U_{1} and Y_{0}Z_{0} is quasi-∼_{w}-reduced. If Y_{0}Z_{0} is ∼_{w}-reduced, we have

x = U_{1}^{−}V_{y}U_{1} ≡ Z_{0}^{−}X_{0}Y_{0}X_{0}X_{0}^{−}Z_{0}
= Z_{0}^{−}X_{0}Y_{0}Z_{0}.

We have V_{x} ≡ X_{0}Y_{0} and V_{y} ≡ Y_{0}X_{0}. Otherwise, i.e. Y_{0}Z_{0} is not ∼_{w}-reduced. There are u, v ∈ G_{i} for some i such that Y_{0} ≡ Y u and Z_{0} ≡ vZ_{1} and Y (uv)Z_{1} is ∼_{w}-reduced. Let X ≡ uX_{0}. Then Y X ≡ V y and

x = Z_{1}^{−}v^{−}X_{0}Y uX_{0}X_{0}^{−}vZ_{1}V = Z_{1}^{−}v^{−}u^{−}uX_{0}Y uvZ_{1}
= ((uv)Z_{1})^{−}XY (uv)Z_{1}.

We have V_{x} ≡ XY and V_{y} ≡ Y X.

If U_{1}^{−}V_{y} is not quasi ∼_{w}-reduced, we have X_{0}, Y_{0}, Z_{0} such that U_{1}^{−} ≡ Z_{0}^{−}Y_{0}^{−}, V_{y} ≡ Y_{0}X_{0}, and Z_{0}^{−}X_{0} is quasi ∼_{w}-reduced. Then we argue as before and have the conclusion. □

Lemma 2.24. Let V be a cyclically ∼-reduced word and W be a ∼- reduced word. If W′^{−}V′W′ be the canonical presentation of W^{−}V W, then W′ is empty, W′ is a tail of W, or W′ ≡ aZ such that V ≡ Y uX, W ≡ X^{−}vZ and a = uv.

Proof. This cooresponds to the case case W_{y} ≡ ⟨⟩ and U_{1} ≡ U_{0} in the proof of Lemma 2.23. When Y_{0}Z_{0} is ∼-reduced, W′ is empty or W′ is a tail of W. Otherwise, Z_{0} ≡ vZ and Y_{0} ≡ Y u and we have the conclusion. □

Lemma 2.25. Let u, v be letters and X^{−}uX be ∼_{w}-reduced for a nonempty word X. Then X^{−}uXv is not conjugate to a single letter.

Proof. Let v ∈ G_{j_{0}}. If the rightmost letter x of X does not belong to G_{j_{0}}, X^{−}uXv is cyclically ∼_{w}-reduced. Otherwise let Y x ≡ X. If x ∈ G_{j_{0}} and xv = e, then X^{−}uXv = x^{−}Y^{−}uY and the right term is cyclically ∼_{w}-reduced. If x ∈ G_{j_{0}} and xv ≠ e, then X^{−}uXv = (xv)^{−}(xvx^{−})Y^{−}uY (xv) and (xvx^{−})Y^{−}uY is cyclically ∼_{w}-reduced. In any case X^{−}uXv is not conjugate to a single letter by Lemma 2.23. □

Lemma 2.26. Let X, Y, Z be words and Y ≠ e. If (XY)^{−} ≡ Y Z, then Y is of the form W^{−}xW with x^{2} = e.

Proof. For each letter x x and x^{−} appear only finitely many times we can see Y ≡ Y^{−}. Then we can easily get the conclusion [3, Corollary 1.6]. □

Lemma 2.27. Let G_{i} ∈ S and V, W ∈ W(⋃_{i∈I} G_{i}) wher V is a cyclically ∼-reduced word which is not a singleton. Then there exist m, n such that both the leftmost V and the rightmost V in V^{m}W V^{n} are stable.

Proof. First we choose m so that the leftmost two copies of V in V^{m}W are stable. Let V V W′ be the ∼_{w}-reduced word of V^{m}W. We claim that the leftmost V in V V W′V^{n} is stable for every n, which suffices to see the conclusion. Suppose that the leftmost V in V V W′V^{n} is not stable for some n. Since the second V from the left in V V W′V^{n} disappears in the ∼_{w}-reduced word, we have words X, Y, Z with Y ≠ e such that V ≡ XY and V^{−} ≡ Y Z by Corollary 2.19 (3). But this does not happen by Lemma 2.26 and the fact that every letter is an element of a torsion-free group. □

Lemma 2.28. Let W ∈ W^{σ}(G_{i}, c_{i} : i ∈ I) for c_{i} ∈ C_{G_{i}}. The ∼_{w}- reduced word of W also belong to W^{σ}(G_{i}, c_{i} : i ∈ I).

Proof. We have C_{i} ∈ c_{i} such that W ∈ W^{σ}(⋃_{i∈I} C_{i}). When C_{i}’s are finite, W takes only finitely many values on such C_{i}’s and there no difficulty for dealing with such C_{i}’s. We assume that each C_{i} is infinite and choose h_{i} : × ×_{n<ω}ℤ_{n} → G_{i} so that Im(h_{i}) = C_{i} and h_{i} is injective, i.e. h(δ_{m}) = h(δ_{n}) implies m = n. We recall the proof of Theorem 2.17 and use the notation there. The ∼_{w}-reduced word of W is obtained from W_{∞} ∈ W^{σ}(⋃_{i∈I} C_{i}). Now for each i ∈ I and α ∈ \overline{W_{∞}} we have U_{α} ∈ W(C_{i}) and u_{α} = U_{α} by Lemma 2.8. Fix i ∈ I. Then {u_{α} : u_{α} ∈ G_{i}} ∈ c_{i} by Lemma 2.16 and we have the conclusion. □

Definition 2.29. Let Ad_{0}, Ad_{1} and Ad_{2}be the subgroups of × ×_{⋃{G_{i}:i∈I}}^{σ}ℤ consisting of elements expressed by words in W_{adm}^{σ}(G_{i} : i ∈ I), W^{σ}(G_{i}, c_{i} : i ∈ I) and W^{fin}(G_{i}, c_{i} : i ∈ I) respectively. Now × ×_{adm}^{σ}(G_{i} : i ∈ I) and × ×^{σ}(G_{i}, c_{i} : i ∈ I) and ∗(G_{i}, c_{i} : i ∈ I) are the quotient group of Ad_{0}, Ad_{1} and Ad_{2} by the equivalence ∼_{w} respectively.

Let h be a homomorphism from ∗_{i∈F} G_{i} to an n-slender group S. For x ∈ × ×_{adm}^{σ}(G_{i} : i ∈ I), let x_{F}^{h} be a member of ∗_{i∈F} G_{i} expressed as U_{F}^{h}, where U is a ∼_{w}-reduced word for x.

Before we investigate these groups more deeply, we state simple results related to the usual free products. The first one is clear by definition.

Proposition 2.30. If c_{i} is the minimal element of C_{G_{i}} for each i ∈ I, × ×^{σ}(G_{i}, c_{i} : i ∈ I) = × ×_{i∈I}^{σ}G_{i} and ∗(G_{i}, c_{i} : i ∈ I) = ∗_{i∈I}G_{i}.

Proposition 2.31. Let G_{0}, G_{1} ∈ S, c_{0} ∈ C_{G_{0}} and c_{1} ∈ C_{G_{1}}. If one of G_{0} and G_{1} is n-slender, then (G_{0}, c_{0}) ∗ (G_{1}, c_{1})(= ∗_{i∈{0,1}}(G_{i}, c_{i})) is the free product G_{0} ∗ G_{1}.

Proof. Suppose that G_{0} is n-slender. Then c_{0} consists of finite subsets of G. Hnece, a word in W^{fin}(G_{i}, c_{i} : i ∈ I) is of the form V_{0}W_{0} ⋯ W_{n}V_{n+1} where V_{i} ∈ W_{adm}^{σ}(G_{0}) and W_{i} ∈ W_{adm}^{σ}(G_{1}). Since V_{i} and W_{i} correspond to elements of G_{0} and G_{1} by Lemma 2.8, we get the conclusion. □

Lemma 2.32. Let G_{i} ∈ S and c_{i} ∈ C_{G_{i}}. Then × ×_{adm}^{σ}(G_{i} : i ∈ I) belongs to S and hence × ×^{σ}(G_{i}, c_{i} : i ∈ I) and ∗(G_{i}, c_{i} : i ∈ I) also belong to S.

Proof. Let u ∈ × ×_{adm}^{σ}(G_{i} : i ∈ I) be not the identity. By definition there exist a word U which presents u, an n-slender group S and a homomorphism h : ∗_{i∈F} G_{i} → S such that h(U_{F}^{h}) ≠ e. Since we can extend this homomorphism h to \overline{h} : Ad_{0} → S so that \overline{h}(W) = e for W ∼_{w} e, \overline{h} factors through × ×_{adm}^{σ}(G_{i} : i ∈ I) and \overline{h}(U) ≠ e and we have the conclusion. □

We introduce a notion and state some basic facts. For x ∈ × ×_{adm}^{σ}(G_{i} : i ∈ I), let L(x) be the set of letters u and u^{−} such that u appears the ∼_{w}-reduced word for x. We remark the following: for a word X = x L(X) is defined as L(x) and hence a letter in X may not belong to L(X); and neither u nor u^{−} may appear in X for u ∈ L(X).

Lemma 2.33. Let X be a ∼_{w}-reduced word and y = W^{−}V W be a canonical presentation. If L(y) ⊊ L(X), then

(1) W is stable in XW^{−}V W, or

(2) W is stable in XW^{−}V^{−}W.

Proof. If L(W^{−}V) ⊊ L(X), the first statement (1) holds by Corollary 2.19. Otherwise, i.e. L(W^{−}V) ⊆ L(X), there exists a letter u in W which does not belong to L(X). Since W^{−}V^{−} is ∼_{w}-reduced and W^{−}V^{−}W is quasi-∼_{w}-reduced, we have the second statement (2). □

Lemma 2.34. Let G_{i} ∈ S and V_{0}, V_{1}, W ∈ W(⋃_{i∈I} G_{i}) where V_{0} and V_{1} are cyclically ∼-reduced words which are not singletons. If L(V_{0}) ≠ L(V_{1}) and V_{0}W is ∼-reduced, then there exist m, n such that both the leftmost V_{0} and the rightmost V_{1} in V_{0}^{m}W V_{1}^{n} are stable.

Proof. First we choose l so that the rightmost two copies of V_{1} in V_{0}W V_{1}^{l} are stable. If L(V_{1}) ⊊ L(V_{0}), the rightmost copy of V_{1} in V_{0}^{m}W V_{1}^{l} is stable for every m < ω and we have the conclusion. Otherwise, a head of V_{0} is stable in V_{0}W V_{1}^{l} or the left most letter in the ∼_{w}-reduced word of V_{0}W V_{1}^{l} belongs to the same group G_{i} as the leftmost letter in V_{0}. Therefore the leftmost V_{0} of V_{0}V_{0}W V_{1}^{l} is stable in V_{0}V_{0}W V_{1}^{l}. □

Remark 2.35. (1) A union of admissible sets may not be admissible. To see this, consider two admissible subsets {δ_{2n} : n < ω} and {δ_{2n+1} : n < ω} of × ×_{n<ω}ℤ_{2n} ∗ × ×_{n<ω}ℤ_{2n+1}. Then, their union {δ_{2n} : n < ω} ∪ {δ_{2n+1} : n < ω} = {δ_{n} : n < ω} is not admissible by [1, Theorem 4.1](see also [2]). This also follows from Corollary 3.20.

(2) For G_{i} ∈ S a word in W^{σ}(G_{i} : i ∈ I) is reduced [3, Definition 1.3] if and only if it is ∼_{w}-reduced.

(3) When the index set I is finite in Definition 2.10, we have W^{σ}(G_{i}, c_{i} : i ∈ I) = W^{fin}(G_{i}, c_{i} : i ∈ I) and consequently × ×^{σ}(G_{i}, c_{i} : i ∈ I) = ∗(G_{i}, c_{i} : i ∈ I).

(4) When every c_{i} is the family of finite subsets of G_{i}, × ×^{σ}(G_{i}, c_{i} : i ∈ I) and ∗(G_{i}, c_{i} : i ∈ I) coincide with the free σ-product × ×_{i∈I}^{σ}G_{i} [3] and the usual free product ∗_{i∈I}G_{i} respectively.

(5) The compatibilities of {C_{0}, C_{1}} and {C_{1}, C_{2}} do not imply the compatibility of {C_{0}, C_{2}} in general. To see this, let C_{0}, C_{1}, C_{2} be pair-wise disjoint countably infinite sets and G be a subgroup of × ×_{i∈C_{0}∪C_{1}∪C_{2}}^{σ}ℤ_{i} generated by × ×_{i∈C_{0}∪C_{1}}^{σ}ℤ_{i} ∪× ×_{i∈C_{1}∪C_{2}}^{σ}ℤ_{i}. Then {C_{0}, C_{1}} and {C_{1}, C_{2}} are compatible in G, but {C_{0}, C_{2}} is not.

(6) We cannot replace Y ∼_{w} ⟨⟩ by “Y = e as a word of W(G_{0} ∪ G_{1}) in the property (2) of Theorem 2.17. We show this by showing an example. Let a_{2n,i} ∈ G_{0}, a_{2n+1,i} ∈ G_{1} be such that a_{n0} = a_{n2}^{−}a_{n1}^{−} for yn < ω. Define \overline{W} = Seq(3) with the lexicographical ordering and W (s) = a_{ls(s),s(ls(s))}. Then W ∼_{w} ⟨⟩ but W satisfies the properties (2) and (3) of Theorem 2.17.

3. Groups × ×^{σ}(G_{i}, c_{i} : i ∈ I) and ∗(G_{i}, c_{i} : i ∈ I)

We shall investigate X_{G} for G = × ×^{σ}(G_{i}, c_{i} : i ∈ I)and ∗(G_{i}, c_{i} : i ∈ I) in the next section. For the purpose we need word theoretic arguments which generalize results in [5, Section 2].

Definition 3.1. Let g be a homomorphism from G_{i} to an n-slender group S. For a word U ∈ W_{adm}^{σ}(G_{i} : i ∈ I), l_{i}^{g}(U) is the cardinality of the set \{α ∈ \overline{U} : U(α) ∈ G_{i}, g(U(α)) ≠ e\}. For u ∈ × ×_{adm}^{σ}(G_{i} : i ∈ I), l_{i}^{g}(u) is l_{i}^{g}(U) where U is the ∼_{w}-reduced word for u.

Lemma 3.2. Let g : G_{i} → S be a homomorphism as in Definition 3.1 and u_{k} ∈ × ×_{adm}^{σ}(G_{i} : i ∈ I) and X_{k} ∈ W_{adm}^{σ}(G_{i} : i ∈ I) be such that

(1) g(u_{k}) ≠ e for each 0 ≤ k ≤ n;

(2) g([X_{k}]_{w}) = e for each 0 ≤ k ≤ n + 1;

(3) each X_{k} is ∼_{w}-reduced for 0 ≤ k ≤ n + 1;

(4) [X_{k}]_{w} ∉ G_{i} for each 1 ≤ k ≤ n.

Then the ∼_{w}-reduced word of X_{0}u_{0}X_{1} ⋯ X_{n}u_{n}X_{n+1} is of the form Y_{0}v_{0}Y_{1} ⋯ Y_{n}v_{n}Y_{n+1} where

(1) g(v_{k}) ≠ e for each 0 ≤ k ≤ n;

(2) g([Y_{k}]_{w}) = e for each 0 ≤ k ≤ n + 1;

(3) each Y_{k} is ∼_{w}-reduced for 0 ≤ k ≤ n + 1.

Proof. When no X_{k} has a letter in G_{i} as the left most letter nor the right most letter for 1 ≤ k ≤ n and the right most letter in X_{0} is not a letter in G_{i} and the left most letter in X_{n+1} is not a letter in G_{i}, if they exist, X_{0}u_{0}X_{1} ⋯ X_{n}u_{n}X_{n+1} is ∼_{w}-reduced by Corollary 2.19(3) and so no proof is necessary. Let X_{0} ≡ Y_{0}x_{0}, X_{k} ≡ x_{2k−1}Y_{k}x_{2k} for 1 ≤ k ≤ n, and X_{n+1} ≡ x_{2n+1}Y_{n+1} where x_{k} ∈ G_{i} or x_{k} is the empty letter and let v_{k} = x_{2k}u_{k}x_{2k+1} for 0 ≤ k ≤ n. Since g(x_{k}) = e, we have g(v_{k}) ≠ e and g([Y_{k}]_{w}) = e. Since x_{2k}, x_{2k+1} ∈ G_{i}, we have [Y_{k}]_{w} ∉ G_{i}. Now v_{k}’s and Y_{k}’s satisfy the desired conditions by the preceding argument. □

Lemma 3.3. Let g : G_{i} → S and others be as in Definition 3.1 and r_{0} : ∗_{i∈F_{0}}G_{i} → T_{0} be a homomorphism to an n-slender group T_{0}. If l_{i}^{g}(u) = l_{i}^{g}(u_{F_{0}}^{r_{0}}) and r_{1} : ∗_{i∈F_{1}}G_{i} → T_{1} be a homomorphism to an nslender group T_{1} such that F_{0} ⊆ F_{1} and r_{1}(x) = e implies r_{0}(x) = e for each x ∈ ∗_{i∈F_{0}}G_{i}, then l_{i}^{g}(u) = l_{i}^{g}(u_{F_{1}}^{r_{1}}).

Proof. Let U be the ∼_{w}-reduced word for u and

U ≡ X_{0}U(α_{0})X_{1} ⋯ X_{n}U(α_{n})X_{n+1}

where l_{i}^{g}(u) = n+1 and \{α_{0}, ⋯, α_{n}\} = \{α ∈ \overline{U} : U(α) ∈ G_{i}, g(U(α)) ≠ e}. Since l_{i}^{g}(u_{F_{0}}^{r_{0}}) = l_{i}^{g}(u) = n + 1,

U_{F_{0}}^{r_{0}} ≡ (X_{0})_{F_{0}}^{r_{0}}U(α_{0})(X_{1})_{F_{0}}^{r_{0}} ⋯ (X_{n})_{F_{0}}^{r_{0}}U(α_{n})(X_{n+1})_{F_{0}}^{r_{0}}.

We remark (X_{k})_{F_{0}}^{r_{0}} ∉ G_{i} for 1 ≤ k ≤ n. By the assumption of r_{1}

U_{F_{1}}^{r_{1}} ≡ (X_{0})_{F_{1}}^{r_{1}}U(α_{0})(X_{1})_{F_{1}}^{r_{1}} ⋯ (X_{n})_{F_{1}}^{r_{1}}U(α_{n})(X_{n+1})_{F_{1}}^{r_{1}}.

Fix 1 ≤ k ≤ n. We have words of finite length U_{0}, ⋯, U_{m} and V_{0}, ⋯, V_{m+1} so that r_{0}(V_{j}) = e for 0 ≤ j ≤ m + 1 and

(X_{k})_{F_{0}}^{r_{0}} ≡ U_{0} ⋯ U_{m}
(X_{k})_{F_{1}}^{r_{1}} ≡ V_{0}U_{0}V_{1} ⋯ V_{m}U_{m}V_{m+1}.

Therefore (X_{k})_{F_{1}}^{r_{1}} ∉ G_{i}. Since g(X_{k}(α)) = e if X_{k}(α) ∈ G_{i}, we conclude l_{i}^{g}(u) = l_{i}^{g}(u_{F_{1}}^{r_{1}}). □

Lemma 3.4. Let H_{j} ∈ S^{σ}, g : H_{j^{∗}} → S be a homomorphism for an n-slender group S and h : × ×_{i<ω}ℤ_{i} → × ×_{adm}^{σ}(H_{j} : j ∈ J) be a homomorphism. Then there exists n < ω such that sup{l_{j^{∗}}^{g}(h(x)) : x ∈ × ×_{i≥n}ℤ_{i}} < ∞.

Proof. Suppose the contrary. We have x_{n} ∈ × ×_{i≥n}ℤ_{i} for n < ω such that sup{l_{j^{∗}}^{g}(h(x_{n})) : n < ω} = ∞. Since we can take a countable subset J′ of J so that j^{∗} ∈ J′ and h(x_{n}) ∈ × ×_{adm}^{σ}(H_{j} : j ∈ J) and there is a projection from × ×_{adm}^{σ}(H_{j} : j ∈ J) to × ×_{adm}^{σ}(H_{j} : j ∈ J′), we may assume J = ω. Let H_{j} ≤ Π_{k<ω}S_{jk} for n-slender groups S_{jk} and p_{jm} : Π_{k<ω}S_{jk} → Π_{k<m}S_{jk} be the projection. We assume S_{j^{∗}0} = S and g is the restriction of p_{j^{∗}1} to H_{j^{∗}}. And let r_{m} = (q_{j} : j < m) where q_{j} : H_{j} → Π_{k<m}S_{jk} is a restriction of the projection p_{jm}. We remark that Π_{k<m}S_{jk} is n-slender. Modifying the proof of [5, Lemma 2.6], we define k_{m} < ω, j_{m} < ω by induction as follows.

Let U_{n} be the ∼_{w}-reduced word for h(x_{n}). Let k_{0} so that l_{j^{∗}}^{g}(h(x_{k_{0}})) > 0 and choose j_{0} < ω so that j_{0} > j^{∗} and l_{j^{∗}}^{g}(h(x_{k_{0}})) = l_{j^{∗}}^{g}((U_{k_{0}})_{j_{0}}^{r_{j_{0}}}) by Lemma 2.14. In the m-th step we choose k_{m} so that 2l_{j^{∗}}^{g}(h(x_{k_{0}} ⋯ x_{k_{m−1}})) < l_{j^{∗}}^{g}(h(x_{k_{m}})) and q_{j_{m−1}}h(× ×_{i≥k_{m}}ℤ_{i}) = {e}. Then we choose j_{m} < ω so that l_{j^{∗}}^{g}(h(x_{k_{0}} ⋯ x_{k_{m}}) = l_{j^{∗}}^{g}((U_{k_{0}} ⋯ U_{k_{m}})_{j_{m}}^{r_{jm}}) by Lemma 2.14. Let W_{m} be a reduced word for x_{k_{m}}. Since (x_{k_{m}} : m < ω) is a proper sequence, W_{0} ⋯ W_{m} ⋯ = W becomes a word naturally. We have j_{m} such that l_{j^{∗}}^{g}(h(W)) = l_{j^{∗}}^{g}(h(W)_{j_{m}}^{r_{jm}}) by Lemmas 2.14 and 3.2. The following inequality follows from Lemma 3.3 and the facts that W_{m+1} ⋯ ∈ × ×_{ω\k_{m+1}}ℤ and W_{m+2} ⋯ ∈ × ×_{ω\k_{m+2}}ℤ:

l_{j^{∗}}^{g}(h(W)) = l_{j^{∗}}^{g}(h(W)_{j_{m}}^{r_{jm}}) = l_{j^{∗}}^{g}(h(W_{0} ⋯ W_{m})_{j_{m}}^{r_{jm}}) = l_{j^{∗}}^{g}(h(W_{0} ⋯ W_{m})) < l_{j^{∗}}^{g}(h(W_{0} ⋯ W_{m}W_{m+1}))

= l_{j^{∗}}^{g}(h(W_{0} ⋯ W_{m}W_{m+1})_{j_{m+1}}^{r_{j_{m+1}}})

= l_{j^{∗}}^{g}(h(W)_{j_{m+1}}^{r_{j_{m+1}}}) = l_{j^{∗}}^{g}(h(W)),

which is a contradiction. □

Lemma 3.5. Let H_{j} ∈ S^{σ}, h : × ×_{n<ω}ℤ_{n} → × ×_{adm}^{σ}(G_{j} : j ∈ J) be a homomorphism, and h(δ_{n}) = W_{n}^{−}V_{n}W_{n} be canonical presentations for n < ω.

Then the same letter does not appear in infinitely many V_{n}’s.

Proof. Suppose that a letter s ∈ H_{j^{∗}} appears in infinitely many V_{n}’s. We have a homomorphism g from H_{j^{∗}} to for an n-slender group S such that g(s) ≠ e. If s appears in infinitely many V_{n}’s which are singletons, then V_{n} = s for such V_{n}’s and gh(δ_{n}) ≠ e for infinitely many n’s, which contradicts the n-slenderness of S. Otherwise, s appears in infinitely many V_{n}’s which are not singletons, then for such a V_{n} V_{n}^{m}’s are ∼- reduced and hence sup{l_{j^{∗}}^{g}(h(δ_{n}^{m})) : m < ω} = ∞, which contradicts Lemm 3.4. □

Definition 3.6. For a sequence (W_{n} : n < ω) of words, W is a taillimit of (W_{n} : n < ω), if the following hold:

(1) For each α ∈ \overline{W} and for all but finite n there exists a word X_{n} such that W_{n} ≡ X_{n}W^{α}, where W^{α} is defined by \overline{W^{α}} = \{β ∈ \overline{W} : β ≥ α\} and W^{α}(β) = W (β) for β ≥ α;

(2) W is maximal among words satisfying (1), i.e. if V satisfies (1), W ≡ XV for some word X.

Lemma 3.7. Let W_{0}^{−}g_{0}W_{0} and W_{1}^{−}g_{1}W_{1} be canonical presentations. Unless W_{0} ≡ W_{1}, the reduced word of W_{0}^{−}g_{0}W_{0}W_{1}^{−}g_{1}W_{1} cannot be of the form W^{−}gW, i.e. a conjugate of a single letter.

Proof. Since W_{0}^{−}g_{0}W_{0} and W_{1}^{−}g_{1}W_{1} are ∼_{w}-reduced, we have ∼_{w}- reduced words U, X_{0}, X_{1} by Corollary 2.18 so that W_{0} ≡ X_{0}U, W_{1} ≡ U^{−}X_{1}, X_{0} ≠ X_{1} and X_{0}X_{1} is quasi-∼_{w}-reduced and is non-empty. If the both X_{0} and X_{1} are non-empty, the ∼_{w}-reduced word of W_{0}^{−}g_{0}W_{0}W_{1}^{−}g_{1}W_{1} is of the form U^{−}X_{0}^{−}g_{0}Y g_{1}X_{1}^{−}U for some Y and we have the conclusion. Since one of X_{0} and X_{1} is non-empty, we deal with the case X_{0} ≡ ⟨⟩. Then W_{0}^{−}g_{0}W_{0}W_{1}^{−}g_{1}W_{1} = U^{−}g_{0}X_{1}g_{1}X_{1}^{−}U. If g_{0}X_{1} is quasi- ∼_{w}-reduced, the ∼_{w}-reduced word of W_{0}^{−}g_{0}W_{0}W_{1}^{−}g_{1}W_{1} is of the form U^{−}Y g_{1}X_{1}^{−}U for some non-empty Y where the leftmost letter in Y is not the inverse of the rightmost letter in X_{1}^{−} and we have the conclusion. The remaining case is when a cancellation occurs in g_{0}X_{1}. Since U^{−}g_{0} is ∼_{w}-reduced, The ∼_{w}-reduced word of W_{0}^{−}g_{0}W_{0}W_{1}^{−}g_{1}W_{1} is of the form U^{−}Y g_{1}Y^{−}g_{0}^{−}U for some Y and we have the conclusion. □

The next lemma is a variant of [5, Lemma 2.8] and its proof is the same. We omit the proof.

Lemma 3.8. Let W_{n} ∈ W_{adm}^{σ}(G_{i} : i ∈ I) (n < ω) be ∼-reduced words. If sup{l_{i^{∗}}^{g}(W_{n}) : n < ω} < ∞ for every n-slender group S and every homomorphism g : G_{i^{∗}} → S, there exists a unique tail-limit of (W_{n} : n < ω) which belongs to W_{adm}^{σ}(G_{i} : i ∈ I) and is ∼-reduced.

Lemma 3.9. Let G_{i} ∈ S and h : × ×_{n<ω}ℤ_{n} → × ×_{adm}^{σ}(G_{i} : i ∈ I) be a homomorphism and h(δ_{n}) = W_{n}^{−}V_{n}W_{n} be canonical presentations for n < ω and W be the tail-limit of W_{n}’s. Define h′ : × ×_{n<ω}ℤ_{n} → × ×_{adm}^{σ}(G_{i} : i ∈ I) by: h′(x) = W h(x)W^{−}. Let h′(δ_{n}) = W′_{n}^{−}V′_{n}W′_{n} be canonical presentations for n < ω. Then the tail-limit of W′_{n}’s is the empty word.

Proof. To show by contradiction, suppose that the tail-limit W′ of W′_{n}’s is not empty. If W is a tail of almost all W_{n}’s, W′_{n} is a head of almost all W_{n}’s, which contradicts the maximality of the tail-limit. Otherwise, there exist infinitely many W_{n}’s such that W is not a tail of W_{n}. Let J be the set of such n’s. If there exist infinitely many n ∈ J such that W_{n} are not tails of W, a tail of W′_{n} contains a tail of W^{−} for such an n. This contradicts that W is the tail-limit of W_{n}’s. In the remaining case W_{n} are tails of W for almost all n ∈ J. For such an n ∈ J let U_{n}W_{n} ≡ W. Then W W_{n}^{−}V_{n}W_{n}W^{−} = U_{n}V_{n}U_{n}^{−}. By Lemma 2.24 W′_{n} is empty, W′_{n} is a tail of U_{n}^{−}, or W′_{n} ≡ a_{n}Z_{n} such that V_{n} ≡ Y_{n}u_{n}X_{n}, U_{n}^{−} ≡ X_{n}^{−}v_{n}Z_{n} and a_{n} = u_{n}v_{n}. Since v_{n}Z_{n} is a tail of W^{−}, a tail of W′ is a tail of W^{−} and consequently some tail T of W′ is a tail of W^{−} and is also a tail of W′_{n} for sufficiently large n ∈ J. Since ⋂_{n∈J} Z_{n} ⊆ ⋂_{n∈J} U_{n}^{−} is the empty word, Z_{n}’s are empty and a_{n}’s are constant for sufficiently large n ∈ J. Since v_{n}’s for sufficiently large n ∈ J are the rightmost letter in W^{−}, v_{n}^{−} belongs to W_{n} but does not belong to U_{n}, which is a contradiction. □

Lemma 3.10. Let G_{i} ∈ S and h : × ×_{n<ω}ℤ_{n} → × ×_{adm}^{σ}(G_{i} : i ∈ I) be a homomorphism and h(δ_{n}) = W_{n}^{−}V_{n}W_{n} ≠ e be canonical presentations and W be the tail-limit of W_{n}’s. For each ascending sequence (n_{k} : k < ω) the tail-limit of W_{n_{k}}’s coincides with W or uW for some letter u.

Proof. Since the tail-limit of W_{n_{k}}’s extends W, we prove that the taillimit of W_{n_{k}}’s does not contain two more letters than W. Suppose that ⋯ u_{0} ⋯ u_{1} ⋯ W be the tail-limit of W_{n_{k}}’s and l_{i}^{g}(u_{0}) ≠ 0 for i with u_{0} ∈ G_{i} and a homomorphism g from G_{i} to an n-slender group S. Suppose that u_{0} ⋯ u_{1} ⋯ W is a tail of W_{n_{k}}. By Lemma 3.4 we have m_{0} and M such that l_{i}^{g}(h(δ_{n})) < M and g ·h(δ_{n}) = e for every n ≥ m_{0}. We have n > max{n_{k}, m_{0}} such that W_{n} has no tail u_{1} ⋯ W. Even if a tail of W_{n_{k}}W_{n}^{−} is not stable and u_{1} in a head of W_{n_{k}} may be calculated, u_{1} is not calcelled. Let U be the ∼_{w}-reduced word of W_{n_{k}}W_{n}^{−}. When V_{n} is a single letter, we have l_{i}^{g}(W_{n_{k}}W_{n}^{−}V_{n}W_{n}W_{n_{k}}^{−}) = l_{i}^{g}(UV_{n}U^{−}) ≥ 2. When V_{n} is not a single letter, at least one of UV_{n} and V_{n}U^{−} is ∼_{w}-reduced. Since l_{i}^{g}(V_{n}) = 0, we have l_{i}^{g}(W_{n_{k}}W_{n}^{−}V_{n}W_{n}W_{n_{k}}^{−}) = l_{i}^{g}(UV_{n}U^{−}) ≥ 2 also in this case. Therefore lim_{m→∞} l_{i}^{g}(h((δ_{n_{k}}δ_{n}))^{m}) = ∞. Since the pair k_{n} and n can be taken to be arbitrary large, it contradicts Lemma 3.4. □

The next lemma easily follows from the compatibilities of c_{i}’s.

Lemma 3.11. Let x_{0}, ⋯, x_{k} ∈ × ×^{σ}(G_{i}, c_{i} : i ∈ I) and W ∈ W_{adm}^{σ}(G_{i} : i ∈ I) such that W ∈ W(⋃_{j=0}^{k} L(x_{j})). Then [W]_{w} ∈ × ×^{σ}(G_{i}, c_{i} : i ∈ I).

To show the next lemma and the main theorem of this section we need to investigete what letters appear in ∼_{w}-reduced words for elements in the image of a homomorphism. We introduce an auxilliary notion for it. A cancellation from UX^{−}XV to UV is a clear cancellation, if UV is ∼_{w}-reduced. (See Corollary 2.19(3).)

Lemma 3.12. Let G_{i} ∈ S and h : × ×_{n<ω}ℤ_{n} → × ×^{σ}(G_{i}, c_{i} : i ∈ I) be a homomorphism. Let h(δ_{n}) = W_{n}^{−}V_{n}W_{n} ≠ e be canonical presentations for n < ω and W be the tail-limit W of W_{n}’s. Then [W]_{w} belongs to × ×^{σ}(G_{i}, c_{i} : i ∈ I).

A similar statement for ∗(G_{i}, c_{i} : i ∈ I) also holds.

Proof. If W is a subword of some W_{n}, then it is obvious. Otherwise, we have n_{k} < n_{k+1} and subwords X_{k}, Y_{k} and U_{k} of W_{n_{k}} such that

(1) X_{k}U_{k} ≡ W_{n_{k}};

(2) U_{k} is a maximal tail of W_{n_{k}} which is a tail of W;

(3) Y_{k+1}U_{k} ≡ U_{k+1} and W = ⋃_{k<ω} U_{k};

(4) Y_{k+1} is non-empty. If V_{n_{k}}’s are single letters for almost all k, using Lemma 3.7 and an endomorphism δ_{n} ↦ δ_{2n}δ_{2n+1} on × ×_{n<ω}ℤ_{n} we reconstruct h so that V_{n_{k}}’s are not single letters and the tail-limit W is unchanged. We divide the argument into two cases. (Case 1): There exist infinitely many W_{n_{k}}’s such that X_{k}’s are nonempty.

By taking a subsequence and using Lemma 3.4, we may assume that n_{k} = k and also the existence of homomorphisms g_{n} : G_{i_{n}} → S_{n} and m_{n} < ω with the following:

• S_{n}’s are n-slender groups;

• l_{i_{n}}^{g_{n}}(V_{n}) ≠ 0;

• sup{l_{i_{n}}^{g_{n}}(h(x)) : x ∈ × ×_{i≥n+1}ℤ_{i}} < m_{n}. We remark that after taking a subsequence W is also the tail-limit of the new sequence and the properties (1)-(4) hold.

Next we observe the ∼_{w}-reduced word of h(δ_{0}^{m_{0}} ⋯ δ_{2n}^{m_{2n}} ⋯) and h(δ_{1}^{m_{1}} ⋯ δ_{2n+1}^{m_{2n+1}} ⋯).

h(δ_{0}^{m_{0}} ⋯ δ_{2n}^{m_{2n}}) = W_{0}^{−}V_{0}^{m_{0}}W_{0}W_{2}^{−}V_{2}^{m_{2}}W_{2} ⋯
W_{2n−2}^{−}V_{2n−2}^{m_{2n−2}}W_{2n−2}W_{2n}^{−}V_{2n}^{m_{2n}}W_{2n}
= W_{0}^{−}V_{0}^{m_{0}}X_{0}Y_{1}^{−}Y_{2}^{−}X_{2}^{−}V_{2}^{m_{2}}X_{2} ⋯

X_{2n−2}^{−}V_{2n−2}^{m_{2n−2}}X_{2n−2}Y_{2n−1}^{−}Y_{2n}^{−}X_{2n}^{−}V_{2n}^{m_{2n}}X_{2n}U_{2n}

The last term of this equation is quasi-∼_{w}-reduced and U_{0}^{−}, Y_{1}^{−}, ⋯, Y_{2n−1}^{−} are stable and so appear in the ∼_{w}-reduced word for h(δ_{0}^{m_{0}} ⋯ δ_{2n}^{m_{2n}}). By the choice of m_{2n} about l_{i_{n}}^{g_{n}}(h(x)) and Corollary 2.19, the leftmost V_{n} is stable and appear in the reduced word for h(δ_{0}^{m_{0}} ⋯ δ_{2n}^{m_{2n}} ⋯). By a similar argument we see that Y_{2}^{−}, ⋯, Y_{2n}^{−} appear in the ∼_{w}-reduced word for h(δ_{1}^{m_{1}} ⋯ δ_{2n+1}^{m_{2n+1}} ⋯). Consequently every letter in W^{−}, i.e. U_{0}^{−}Y_{1}^{−}Y_{2}^{−} ⋯ Y_{n}^{−} ⋯, appears in the reduced word for h(δ_{0}^{m_{0}} ⋯ δ_{2n}^{m_{2n}} ⋯) or h(δ_{1}^{m_{1}} ⋯ δ_{2n+1}^{m_{2n+1}} ⋯). By the compatibilities of c_{i}’s we have [W]_{w} belongs to × ×^{σ}(G_{i}, c_{i} : i ∈ I). (Case 2): Otherwise. Then X_{k}’s are empty for almost all k’s and so W_{n_{k}} is a tail of W_{n_{k+1}} for such k. As in Case 1 we assume n_{k} = k also the existence of homomorphisms g_{n} : G_{i_{n}} → S_{n} with the following:

• S_{n}’s are n-slender groups;

• l_{i_{n}}^{g_{n}}(V_{n}) ≠ 0;

• sup{l_{i_{n}}^{g_{n}}(h(x)) : x ∈ × ×_{i≥n+1}ℤ_{i}} < ∞. Since W_{n}^{−}V_{n}^{−} is ∼_{w}-reduced, we use h(δ_{n}^{−}) instead of h(δ_{n}). Next we choose m_{n} < ω by induction so that l_{i_{n}}^{g_{n}}(h(δ_{0}^{−2m_{0}−1} ⋯ δ_{n−1}^{−2m_{n−1}−1})) < m_{n} and sup{l_{i_{n}}^{g_{n}}(h(x)) : x ∈ × ×_{i≥n+1}ℤ_{i}} < m_{n}.

We observe the ∼_{w}-reduced word of h(δ_{0}^{−2m_{0}−1} ⋯ δ_{n}^{−2m_{n}−1} ⋯). By the construction of m_{n}’s m_{n} + 1-th copy of V_{n}^{−} is stable. We recall Y_{n}W_{n} ≡ W_{n+1}. Then there exists ∼_{w}-reduced words Z_{n} such that

h(δ_{0}^{−2m_{0}−1} ⋯ δ_{n}^{−2m_{n}−1} ⋯)
= W_{0}^{−}V_{0}^{−2m_{0}−1}Y_{1}^{−}V_{1}^{−2m_{1}−1} ⋯ Y_{n}^{−}V_{n}^{−2m_{n}−1}Y_{n+1}^{−} ⋯
= Z_{0}V_{0}^{−m_{0}}Z_{1}V_{1}^{−}Z_{2} ⋯ Z_{n}V_{n}^{−}Z_{n+1} ⋯

where the last term of this equation is ∼_{w}-reduced. If for each letter x in W x itself or x^{−} appears in the third term

Z_{0}V_{0}^{−m_{0}}Z_{1}V_{1}^{−}Z_{2} ⋯ Z_{n}V_{n}^{−}Z_{n+1} ⋯

we have the conclusion. Since W ≡ ⋯ Y_{n+1}Y_{n} ⋯ Y_{1}W_{0} and cancellations may occur in the subwords of the second term as V_{n}^{−m_{n}}Y_{n+1}^{−}V_{n+1}^{−m_{n+1}}, what we should take care is the case when a letter in Y_{n} changes by multiplications in some group G_{i}.

We analyze the patterns of V_{n−1}^{−m_{n−1}−1}Y_{n}^{−}V_{n}^{−m_{n}−1} according to cancellations and multiplications concerning letters in Y_{n}^{−}. Observe that Y_{n}^{−}V_{n}^{−m_{n}−1} is ∼_{w}-reduced, then we have the following three patterns:

(0) V_{n−1}^{−}A_{n}B_{n}B_{n}^{−}D_{n}V_{n}^{−} where Y_{n}^{−} is a head of B_{n}^{−};

(1) V_{n−1}A_{n}B_{n}B_{n}^{−}D_{n}V_{n}^{−m_{n}−1} where Y_{n} ≡ B_{n}^{−}D_{n} and V_{n−1}A_{n}D_{n}V_{n}^{−m_{n}−1} is ∼_{w}-reduced;

(2) V_{n−1}A_{n}a_{n}B_{n}B_{n}^{−}b_{n}D_{n}V_{n}^{−m_{n}−1} where Y_{n} ≡ b_{n}B_{n}^{−}D_{n} and a_{n}, b_{n} ∈

G_{i} for some i and a_{n}b_{n} ≠ e. When a cancellation is a clear chancellation, we have (0) or (1) and, otherwise (2). We partition ω so that n ∈ J_{k} if and only if Case(k) happens for 0 ≤ k ≤ 2.

If J_{2} is finite, that is, almost all cancellations are clear ones, then x or x^{−} appears in the third term Z_{0}V_{0}^{m_{0}}Z_{1}V_{1}Z_{2} ⋯ Z_{n}V_{n}Z_{n+1} ⋯ for almost all letter x in the W, which implies the conclusion. We should take care of a_{n}, b_{n}’s in J_{2} when J_{2} is infinite. Since {a_{n}b_{n} : n ∈ J_{2}, a_{n}, b_{n} ∈ G_{i}} ∈ c_{i} and {a_{n} : n ∈ J_{2}, a_{n} ∈ G_{i}} ∈ c_{i}, {b_{n} : n ∈ J_{2}, b_{n} ∈ G_{i}} ∈ c_{i} by Lemma 2.5. Now we have the conclusion by Lemma 3.11.

Since our discussion is closed in W^{fin}(G_{i}, c_{i} : i ∈ I) when we start from words in W^{fin}(G_{i}, c_{i} : i ∈ I), we can see the second statement easily. □

We generalize the notion of the proper sequence for a sequence (x_{n} : n < ω) of elements in × ×_{adm}^{σ}(G_{i} : i ∈ I). A sequence (x_{n} : n < ω) is proper, if lim_{n→∞} l_{i}^{g}(x_{n}) = 0 for every homomorphism g from G_{i} to every n-slender group S. More exactly, our notion coincides with that of [5, Definition 2.1], when G_{i} is n-slender and a sequence is countable.

The next notion is also a generalization of “standard homomorphism” introduced in [5, Definition 2.7] and so we use the same name.

Definition 3.13. For groups G_{i} (i ∈ I), a homomorphism h : × ×_{n<ω}ℤ_{n} → × × _{adm}^{σ}(G_{i} : i ∈ I) is standard, if (h(δ_{n}) : n < ω) is proper and h(W) = V for a word W ∈ W(ℤ_{n} : n < ω), where V is the word in W_{adm}^{σ}(G_{i} : i ∈ I) defined as follows:

(1) \overline{V} = \{(α, β) : α ∈ \overline{W} , β ∈ \overline{V_{α}}\}, where V_{α} is the reduced word of h(W (α));

(2) The order (α, β) < (α′, β′) is lexicographical, i.e. α < α′, or α = α′ and β < β′;

(3) V (α, β) = V_{α}(β) for (α, β) ∈ \overline{V} . For homomorphisms h, h′ : G_{0} → G_{1}, h is conjugate to h′, if there exists g ∈ G′ such that h(x) = g^{−}\overline{h}(x)g for all x ∈ G_{0}.

The proof of the next lemma is straightforward and we omit it.

Lemma 3.14. Let x_{n} = U_{0n} ⋯ U_{kn} for each n < ω and there are W_{0}, ⋯, W_{k} ∈ W_{adm}^{σ}(G_{i} : i ∈ I) such that U_{in}’s are subwords of W_{i} and \overline{U_{im}} ∩ \overline{U_{in}} = ∅ (m ≠ n) for each i. Then (x_{n} : n < ω) is a proper sequence.

Lemma 3.15. Let G_{i} ∈ S for each i ∈ I and h : × ×_{n<ω}ℤ_{n} → × ×_{adm}^{σ}(G_{i} : i ∈ I) be a homomorphism. If (h(δ_{n}) : n < ω) is proper, then h is a standard homomorphism. Consequently, this statement holds for × ×^{σ}(G_{i}, c_{i} : i ∈ I) instead of × ×_{adm}^{σ}(G_{i} : i ∈ I).

Proof. Let W ∈ W(ℤ_{n} : n < ω) and V be the word defined in Definition 3.13. Let g : ∗_{i∈F} G_{i} → S be a homomorphism for F ⋐ I and an nslender group S. By Lemma 2.32 it suffices to show g(h(W)_{F}^{g}) = g(V_{F}^{g}).

Since (h(δ_{n}) : n < ω) is proper, there exist only finitely many α ∈ \overline{W} such that g ∘ l_{i}^{g}(h(W (α)) ≠ 0 for every i ∈ F. Picking up these as w_{0}, ⋯, w_{n}, we let W ≡ W_{0}w_{0}W_{1} ⋯ W_{n}w_{n}W_{n+1}. Then g ∘ h(W_{i}) = e for 0 ≤ 1 ≤ n + 1 and V_{F}^{g} = h(w_{0})_{F}^{g} ⋯ h(w_{n})_{F}^{g} and hence g(h(W)_{F}^{g}) = g(h(w_{0})_{F}^{g} ⋯ h(w_{n})_{F}^{g}) = g(V_{F}^{g}). □

Lemma 3.16. Let G_{i} ∈ S for each i ∈ I and h : × ×_{n<ω}ℤ_{n} → × ×^{σ}(G_{i}, c_{i} : i ∈ I) be a homomorphism. Suppose that there are a proper sequence (u_{n} : n < ω) in × ×_{n<ω}ℤ_{n} and a proper sequence (a_{n} : n < ω) in × ×^{σ}(G_{i}, c_{i} : i ∈ I) such that:

(1) L(h(δ_{n})) ⊆ L(h(u_{n})) ∪ {a_{n}} for each n;

(2) (h(u_{n}) : n < ω) is a proper sequence in × ×^{σ}(G_{i}, c_{i} : i ∈ I);

(3) ⋃_{n<ω} L(h(u_{n})) ∪ {a_{n} : n < ω} ⊆ C ∈ ⋓(c_{i} : i ∈ I). Then h is a standard homomorphism and Im(h) ⊆ H(C) = {[W]_{w} : W ∈ W^{σ}(C)}.

Proof. The properness of a sequence (h(δ_{n}) : n < ω) follows from (1) and (2). By Proposition 1.3, h coincides with a standard homomorphism and hence Im(h) ⊆ H(C) = {[W]_{w} : W ∈ W^{σ}(C)}. □

The next lemma is straightforward and we omit the proof.

Lemma 3.17. Let G_{i} ∈ S and h_{0}, h_{1} be homomorphisms from × ×_{n<ω}ℤ_{n} to × ×^{σ}(G_{i}, c_{i} : i ∈ I). Suppose that h_{1}(δ_{2n}) = h_{0}(δ_{n}) and h_{1}(δ_{2n+1}) = h_{0}(u_{n}) for some proper sequence (u_{n} : n < ω) in × ×_{n<ω}ℤ_{n}. If h_{1} is a standard homomorphism, then h_{0} is also standard and Im(h_{0}) ⊆ Im(h_{1}).

The next lemma is a direct consequence of Lemma 2.5 and we omit its proof.

Lemma 3.18. Let c_{i} ∈ C_{G_{i}} and {x_{n} : n < ω}, {y_{n} : n < ω} ∈ ⋓(c_{i} : i ∈ I). Then {x_{n}^{ε_{n}} : n < ω} ∈ ⋓(c_{i} : i ∈ I) for any ε_{n} = ±1. If {n < ω : x_{n} = x} and {n < ω : y_{n} = x} are finite for x ≠ e, then {x_{n}y_{n} : n < ω} ∈ ⋓(c_{i} : i ∈ I).

We state the main theorem in this section, which is important for the next section and its proof is a modification of that of [5, Lemma 2.8] and uses Lemmas 3.4 and 3.12 instead of [5, Lemmas 2.6 and 2.8].

Theorem 3.19. Let G_{i} ∈ S and h : × ×_{n<ω}ℤ_{n} → × ×^{σ}(G_{i}, c_{i} : i ∈ I) be a homomorphism. Then one of the following holds:

(1) h is conjugate to a homomorphism \overline{h} such that there exists G_{i} such that \overline{h}(δ_{n}) ∈ G_{i} for almost all n < ω;

(2) there exists C ∈ ⋓(c_{i} : i ∈ I) and a standard homomorphism \overline{h} such that Im(\overline{h}) ⊆ H(C) = \{[W]_{w} : w ∈ W^{σ}(C)\} and h is a conjugate to \overline{h}.

Before proceeding to a proof, we state its corollary.

Corollary 3.20. Let G_{i} ∈ S and h : × ×_{n<ω}ℤ_{n} → ∗(G_{i}, c_{i} : i ∈ I) be a homomorphism. Then one of the following holds:

(1’) h is conjugate to a homomorphism \overline{h} such that there exists G_{i}

such that \overline{h}(δ_{n}) ∈ G_{i} for almost all n < ω; (2’) there exists C ∈ ⋓^{F}(c_{i} : i ∈ I) and a standard homomorphism

\overline{h} such that Im(\overline{h}) ⊆ H(C) = \{[W]_{w} : w ∈ W^{σ}(C)\} and h is a

conjugate to \overline{h}.

Proof. Since ∗(G_{i}, c_{i} : i ∈ I) is a subgroup of × ×^{σ}(G_{i}, c_{i} : i ∈ I), we have h(x) = u^{−1}\overline{h}(x)u by Theorem 3.19. Here u is obtained as a tail-limit of conjugators, we have u ∈ ∗(G_{i}, c_{i} : i ∈ I) by Lemma 3.12 and hence \overline{h} : × ×_{n<ω}ℤ_{n} → ∗(G_{i}, c_{i} : i ∈ I), which implies the conclusion. □

We remark that Theorem 3.19 holds, if the theorem holds for a restriction h|× ×_{n≥m}ℤ_{n}, and hence we may disregard conditions of finitely many h(δ_{n})’s. Under the assumption of Lemma 3.21 h said to be with the empty tail-limit, if the tail-limit of W_{n}’s is the empty word.

Lemma 3.21. Theorem 3.19 holds under the following condition where h(δ_{n}) = W_{n}^{−}V_{n}W_{n} is a canonical presentation:

(1) h(δ_{n}) ≠ e for every n < ω;

(2) h is with the empty tail-limit;

(3) there are infinitely many n’s for which V_{n}’s are not single letters.

Proof. We divide our argument into the following four cases according to W_{n}’s. (Remark that one of the cases holds.) Since the tail-limit of W_{n}’s is the empty word, only one of the four cases hold. Let J_{0} = {n < ω : V_{n} is not a single letter }, which is infinite. (Case 0): W_{n}’s are empty for infinitely many n ∈ J_{0}.

We may assume that W_{0} is empty and 0 ∈ J_{0}. We define n_{k} < n_{k+1} inductively as follows. Let n_{0} = 0. Suppose that we have defined n_{k−1}. Let n_{k} be the least n > n_{k−1} such that n ∈ J_{0} and L(h(δ_{n})) ≠ L(h(δ_{n_{k−1}}). Since non-empty L(h(δ_{n}))’s cannot be equal for infinitely many n’s by Lemma 3.4, we can find n_{k}. We define an integer m_{i} > 0 for each i by induction as follows. Since the 0-th step is the n_{0}-th step, we start from the n_{k}-step.

(n_{k}-th step): By Lemm 2.34 we have m such that the leftmost V_{n_{k}} and the rightmost V_{n_{k+1}} in V_{n_{k}}^{m}V_{n_{k+1}}^{m} are stable. Next take g_{k} and j_{k} so that l_{j_{k}}^{g_{k}}(V_{n_{k}}) ≠ 0. Then by Lemma 3.4 for sufficiently large p sup{l_{j_{k}}^{g_{k}}(h(x)) : x ∈ × ×_{i≥p}ℤ_{i}} < ∞. Choose m_{n_{k}} so that m_{n_{k}} > m and m_{n_{k}} > lim_{p→∞} sup{l_{j_{k}}^{g_{k}}(h(x)) : x ∈ × ×_{i≥p}ℤ_{i}}.

(i-th step for n_{k−1} < i < n_{k} ): By Lemma 2.27 we have m_{i} such that both the leftmost V_{n_{k}} and the rightmost one in V_{n_{k}}^{m_{i}}W_{i}^{−}V_{i}W_{i}V_{n_{k}}^{m_{i}} are stable.

We claim: the ∼_{w}-reduced word U_{∞} for

h(δ_{0}^{2m_{0}}δ_{n_{1}}^{m_{0}}δ_{n_{1}}^{m_{1}}δ_{1}δ_{n_{1}}^{m_{1}} ⋯ δ_{n_{k}}^{m_{i}}δ_{i}δ_{n_{k}}^{m_{i}} ⋯ δ_{n_{k}}^{m_{n_{k}−1}}δ_{n_{k}}^{2m_{n_{k}}} δ_{n_{k+1}}^{m_{n_{k}}} ⋯)

is of the form

V_{0}^{m_{0}+1}X_{0}V_{n_{1}}^{2}X_{1}V_{1} ⋯ V_{n_{1}}^{m_{1}+1}X_{n_{1}}V_{n_{2}} ⋯ V_{n_{k}}^{2}X_{i}V_{n_{k}}^{2} ⋯ V_{n_{k}}^{m_{n_{k}} +1}X_{n_{k}}V_{n_{k+1}}^{2} ⋯ X

for some X_{i}’s and X. To see that V_{n_{k−1}}^{m_{n_{k−1}}+1} appear in U_{∞}, we recall the choice of m_{n_{k}}. Then we have p such that sup{l_{j_{k}}^{g_{k}}(h(x)) : x ∈ × ×_{i≥p}ℤ_{i}} < m_{n_{k}}. Now the reduced word for

h(δ_{0}^{2m_{0}}δ_{n_{1}}^{m_{0}} ⋯ δ_{n_{k−1}}^{2m_{k−1}}δ_{n_{k−1}}^{m_{k−1+1}} ⋯ δ_{n_{k}}^{2m_{n_{k}}} δ_{n_{k}}^{m_{n_{k}+1}} ⋯ δ_{n_{l}}^{m_{p}}δ_{p}δ_{n_{l}}^{m_{p}})

is of the form

V_{0}^{m_{0}+1}X_{0}V_{n_{1}}^{2}X_{1}V_{1} ⋯ V_{n_{k}}^{m_{k}+1}X_{n_{k}}V_{n_{k+1}} ⋯ V_{n_{l}}^{2}X_{p}V_{n_{l}} ≡ U_{p}

Since the leftmost V_{n_{k}} remains in a possible cancellation between U_{p} and the reduced word for h(δ_{n_{l}}^{m_{p+1}}δ_{p+1}δ_{n_{l}}^{m_{p+1}} ⋯), V_{n_{k−1}}^{m_{n_{k−1}}+1} remains in U_{∞}. Now we have shown the claim.

Next we think of V_{i}W_{i} for n_{k−1} < i < n_{k}. The ∼_{w}-reduced word of V_{n_{k}}^{m_{i}}W_{i}^{−}V_{i}W_{i}V_{n_{k}}^{m_{i}} appear in U_{∞}. Let Y_{i} be the ∼_{w}-reduced word of W_{i}^{−}V_{i}W_{i} (there may a multiplication in W_{i}^{−}V_{i}). We invetigate the reduction of V_{n_{k}}^{m_{i}}Y_{i}V_{n_{k}}^{m_{i}} precisely. If the multiplications do not concern with letters in Y_{i}, a letter x in Y_{i} is stable or its inverse x^{−} appears in V_{n_{k}}. Let I_{0} be the set of such i’s. When multiplications concern with letters in Y_{i}, i.e. i ∉ {n_{k} : k ∈ ω} ∪ I_{0}, we have the following patterns:

(1) Sy_{0i}Y′_{i}y_{1i}T ≡ Y_{i}, and v_{0i}S^{−} is a tail and T^{−}v_{1i} is a head of V_{n_{k}}^{m_{i}} and each of pairs v_{0i}, y_{0i} and y_{1i}, v_{1i} belong to the same group;

(2) SY′_{i}y_{1i}T ≡ Y_{i}, and S^{−} is a tail and T^{−}v_{1i} is a head of V_{n_{k}}^{m_{i}} and y_{1i}, v_{1i} belong to the same group;

(3) Sy_{0i}Y′_{i}T ≡ Y_{i}, and v_{0i}S^{−} is a tail and T^{−} is a head of V_{n_{k}}^{m_{i}} and v_{0i}, y_{0i} belong to the same group;

(4) Sy_{0i}T ≡ Y_{i}, and v_{0i}S^{−} is a tail and T^{−}v_{1i} is a head of V_{n_{k}}^{m_{i}} and v_{0i}, y_{0i}, v_{1i} belong to the same group.

Let K_{k} be the set of i’s such that the above (k) holds. We only treat with the cases (2) and (4). Since the letter y_{1i}v_{1i}, that is, the elements of groups but not words y_{1i}v_{1i} of two letters, appear in U_{∞}, (y_{1i}v_{1i} : i ∈ K_{2}) is a proper sequence. On the other hand v_{1i}’s appear in U_{∞} since each v_{1i} is a letter in V_{n_{k}}, and hence (v_{1i} : i ∈ K_{2}) is a proper sequence. Then (y_{1i} : i ∈ K_{2}) is a proper sequence and {y_{1i} : i ∈ K_{2}} ∈ ⋃(c_{i} : i ∈ I) by Lemma 3.18. The case (4) is also similar. Though v_{0i}y_{0i}v_{1i} may be the identity, (v_{0i}y_{0i}v_{1i} : i ∈ K_{4}) is a proper sequence. Since v_{0i}’s and v_{1i}’s appear in U_{∞}, (v_{0i} : i ∈ K_{4}) and (v_{1i} : i ∈ K_{4}) are proper sequences and hence (y_{0i} : i ∈ K_{4}) is also a proper sequence and {y_{0i} : i ∈ K_{4}} ∈ ⋃(c_{i} : i ∈ I) by Lemma 3.18.

By the compatiblity of c_{i} we have a countable C ∈ ⋃(c_{i} : i ∈ I) such that Im(h) ⊆ H(C) and h is a standard homomorphism by Lemma 3.16.

(Case 1): Case 0 does not hold and for each n there exists m > n such that W_{m} is non-empty and W_{n}W_{m}^{−} is ∼_{w}-reduced.

We may assume that W_{i} is non-empty if V_{i} is not a single letter. We define n_{k} < n_{k+1} inductively as follows. Let n_{0} be the least n such that W_{n} is non-empty and there exists m > n such that W_{m} is non-empty and W_{n}W_{m}^{−} is ∼_{w}-reduced. Suppose that we have defined n_{k−1}. Let n_{k} be the least n > n_{k−1} such that W_{n} is non-empty and W_{n_{k−1}}W_{n}^{−} is ∼_{w}-reduced. Observe that W_{i}W_{n_{k}}^{−} is ∼_{w}-reduced for n_{k−1} ≤ i < n_{k}. For n_{k−1} ≤ i < n_{k}, let u_{i} = δ_{i}δ_{n_{k}} if W_{i} is empty (remark that V_{i} is a single letter) and u_{i} = δ_{i}^{2}δ_{n_{k}} if W_{i} is non-empty. Then the ∼_{w}-reduced word for h(u_{i}) is cyclically ∼_{w}-reduced and is not a single letter. Let φ be an endomorphism on × ×_{n<ω}ℤ_{n}} satisfying φ(δ_{i}) = u_{i} and apply Case 0 to h · φ. Since L(h(δ_{i})) ⊆ L(h(u_{i})) for each i, we have the conclusion by Lemma 3.16.

(Case 2): Neither Case 0 nor Case 1 holds, and there exists i_{0} such that W_{i_{0}} is non-empty and there’s no maximal element in \overline{W_{i_{0}}} and W_{i_{0}}W_{n}^{−} is not ∼_{w}-reduced for any non-empty W_{n} with n > i_{0}.

Since it is not in Case 0, there are infinitely many non-empty W_{n}’s and we may assume that if W_{i} is empty, V_{i} is a single letter.

(Subcase 2.1): There are infinitely many n’s such that W_{n} are not empty and W_{n} are not tails of W_{i_{0}}.

Since the tail-limit of W_{n}’s is empty and any tail of W_{i_{0}} is not a single letter, for any tail T of W_{i_{0}} there are at most only finitely many i’s such that T is a tail of W_{i} by Lemma 3.10. Disregarding finitely many ℤ_{n}’s we can choose n_{k−1} < n_{k} so that

(a) n_{0} = i_{0} = 0;

(b) W_{n_{k}} is not a tail of W_{i_{0}} and W_{i_{0}} is not a tail of W_{n_{k}} for k ≥ 1;

(c) for each n_{k−1} ≤ i < n_{k}, if W_{i} is non-empty, a head and a tail of W_{i}W_{n_{k}}^{−} remains in the ∼_{w}-reduced word.

We define m_{i} > 0 as in Case 0, but it is simpler according to the last property of W_{i}W_{n_{k}}^{−}. If V_{i} is a single letter, let m_{i} = 1 and otherwise, choose g_{i} and j_{i} so that l_{j_{i}}^{g_{i}}(V_{i}) > 0 and let m_{i} > lim_{p→∞} sup{l_{j_{i}}^{g_{i}}(h(x)) : x ∈ × ×_{n≥p}ℤ_{n}}.

Then we consider the ∼_{w}-reduced word for h(δ_{n_{k}}δ_{i}^{m_{i}}δ_{n_{k}}). If W_{i} is empty, V_{i} is a single letter and so W_{n_{k}}V_{i}W_{n_{k}}^{−} is ∼_{w}-reduced, since \overline{W_{n_{k}}} has no maximal element. Otherwise, a head and a tail of W_{n_{k}}W_{i}^{−}V_{i}W_{i}W_{n_{k}}^{−} and also V_{i} remain in the ∼_{w}-reduced word. Considering the ∼_{w}- reduced word for

h(δ_{0}^{m_{0}}δ_{n_{1}} ⋯ δ_{n_{k−1}}^{m_{n_{k−1}}}δ_{n_{k}} ⋯ δ_{i}^{m_{i}}δ_{n_{k}} ⋯ δ_{n_{k}−1}^{m_{n_{k}−1}}δ_{n_{k}}δ_{n_{k}}^{m_{n_{k}}} δ_{n_{k+1}} ⋯),

we have the conclusion by a similar but simpler argument as in Case 0. (Subcase 2.2): For almost all n W_{n}’s are empty or tails of W_{i_{0}}.

We may assume that i_{0} = 0 and W_{n} is empty or a tail of W_{0} for every n. Using Lemma 3.5 we can choose n_{k−1} < n_{k} so that

(a) n_{0} = i_{0} = 0;

(b) for i < n_{k} W_{n_{k}} is a tail of non-empty W_{i} and W_{i} is not a tail of W_{n_{k}} ;

(c) L(V_{n_{k−1}}) ≠ L(V_{k}) for k < ω. (We remark that a tail always is a non-empty word.)

Let X_{i} be a head of W_{i} such that W_{i} ≡ X_{i}W_{n_{k}} for n_{k−1} ≤ i < n_{k}. Then we define m_{i} similarly to Case 0.

(n_{k}-th step): By Lemma 2.34 we have m such that the leftmost V_{n_{k}} and the rightmost V_{n_{k+1}} in V_{n_{k}}^{m}V_{n_{k+1}}^{m} are stable. Next take g_{k} and j_{k} so that l_{j_{k}}^{g_{k}}(V_{n_{k}}) ≠ 0. Then by Lemma 3.4 for sufficiently large p sup{l_{j_{k}}^{g_{k}}(h(x)) : x ∈ × ×_{i≥p}ℤ_{i}} < ∞. Choose m_{n_{k}} so that m_{n_{k}} > m and m_{n_{k}} > lim_{p→∞} sup{l_{j_{k}}^{g_{k}}(h(x)) : x ∈ × ×_{i≥p}ℤ_{i}}.

(i-th step for n_{k−1} < i < n_{k} ): By Lemma 2.27 we have m_{i} such that both the leftmost V_{n_{k}} and the rightmost one in V_{n_{k}}^{m_{i}}X_{i}^{−}V_{i}X_{i}V_{n_{k}}^{m_{i}} are stable. Then we consider the ∼_{w}-reduced word for

h(δ_{0}^{m_{0}}δ_{n_{1}}^{m_{0}}δ_{n_{1}}^{m_{1}}δ_{1}δ_{n_{1}}^{m_{1}} ⋯ δ_{n_{k}}^{m_{i}}δ_{i}δ_{n_{k}}^{m_{i}} ⋯ δ_{n_{k}}^{m_{n_{k}−1}}δ_{n_{k}}^{m_{n_{k}}} δ_{n_{k+1}}^{m_{n_{k}}} ⋯)

and have the conclusion as in Case 0. (Case 3): Neither Case 0 nor Case 1 holds, and there exists i_{0} such that there’s a maximal element in \overline{W_{i_{0}}} and W_{i_{0}}W_{n}^{−} is not ∼_{w}-reduced when W_{i} is non-empty and i > i_{0}.

Let J_{1} = {i > i_{0} : W_{i} is non-empty} and let W_{i} ≡ U_{i}w_{i} for i ∈ J_{1}. Since Case 0 does not hold, the set J_{1} is infinite and we assume that i ∈ J_{1} if V_{i} is not a single letter. (Subcase 3.1): There exists w such that w = w_{i} for almost all i ∈ J_{1}.

Since the tail-limit of W_{i}’s is empty, there exist infinitely many empty W_{i}’s and all V_{i}’s are single letters for such n’s. Let i^{∗} ∈ I such that w ∈ G_{i^{∗}} and let x be a single letter not in G_{i^{∗}}. If i ∈ J_{1} and w = w_{i}, then w_{i}^{−}U_{i}^{−}V_{i}U_{i}w_{i}x is cyclically ∼_{w}-reduced. Hence, if there are infinitely many i’s such that V_{i}’s are single letters not in G_{i^{∗}}, then we have a contradiction by Lemma 3.5. So for almost all i ∉ J_{1} V_{i} ∈ G_{i^{∗}} and wV_{i}w^{−} ∈ G_{i^{∗}}. We have i_{n} < i_{n+1} and k_{n} < k_{n+1} such that i_{n} ∈ J_{1} and each W_{k_{n}} is empty and V_{k_{n}} is a single letter v_{k_{n}} and ω\{i_{n}, k_{n} : n < ω} is finite. Define \overline{h}(x) = wh(x)w^{−}. Since canonical presentations of \overline{h}(δ_{i}) for i ∉ J_{1} consist of single letters, \overline{h} is also with the empty tail-limit and satisfies the assumption of this lemma, i.e. Lemma 3.21.

Now we have

\overline{h}(δ_{i_{n}}δ_{k_{n}}) = ww^{−}U_{i_{n}}^{−}V_{i_{n}}U_{i_{n}}wv_{k_{n}}w^{−} = U_{i_{n}}^{−}V_{i_{n}}U_{i_{n}}wv_{k_{n}}w^{−}.

Since v_{k_{n}} ∈ G_{i^{∗}} for almost all n’s, U_{i_{n}}V_{i_{n}}U_{i_{n}}(wv_{k_{n}}w^{−}) is cyclically ∼_{w}- reduced and not single letters for almost all n’s. We apply Case 0 to \overline{h} and have the conclusion by Lemma 3.17

(Subcase 3.2): For each i ∈ J_{1} there exists i′ ∈ J_{1} such that i′ > i and w_{i′} ≠ w_{i}.

Since W_{i_{0}}^{−}W_{i} is not ∼_{w}-reduced for i ∈ J_{1}, w_{i} belongs to the same group G_{i^{∗}} to which w_{i_{0}} belongs. We may assume i_{0} = 0 = n_{0} and choose n_{k} < n_{k+1} so that n_{k} ∈ J_{1}, w_{n_{k}} ≠ w_{n_{k+1}} and w_{i} = w_{n_{k−1}} if i ∈ J_{1} and n_{k−1} ≤ i < n_{k}. We define m_{i} = 1 if V_{i} is a single letter. If V_{i} is not a single letter, we define g_{i} and j_{i} so that l_{j_{i}}^{g_{i}}(V_{i}) ≠ 0 and choose m_{i} > 0 as in Subcase 2.1. We have

h(δ_{0}^{m_{0}}δ_{n_{1}}δ_{1}^{m_{1}}δ_{n_{1}} ⋯ δ_{n_{1}−1}^{m_{n_{1}−1}}δ_{n_{1}}δ_{n_{1}}^{m_{n_{1}}}δ_{n_{2}} ⋯ δ_{n_{k}−1}^{m_{n_{k}−1}}δ_{n_{k}}δ_{n_{k}}^{m_{n_{k}}} δ_{n_{k+1}} ⋯)

= W_{0}^{−}V_{0}^{m_{0}}U_{0}w_{0}w_{n_{1}}U_{n_{1}}^{−}V_{n_{1}}U_{n_{1}}w_{n_{1}}w_{1}^{−}U_{1}^{−}V_{1}^{m_{1}}U_{1}w_{1} ⋯

U_{n_{1}−1}^{−}V_{n_{1}−1}^{m_{n_{1}−1}}U_{n_{1}−1}w_{n_{1}−1}w_{n_{1}}^{−}U_{n_{1}}^{−}V_{n_{1}}^{m_{n_{1}}}U_{n_{1}}w_{n_{1}}w_{n_{2}}^{−}U_{n_{2}}^{−}V_{n_{2}}W w_{n_{2}} ⋯

w_{n_{k}−1}^{−}U_{n_{k}−1}^{−}V_{n_{k}−1}^{m_{n_{k}−1}}U_{n_{k}−1}w_{n_{k}−1}w_{n_{k}}^{−}
U_{n_{k}}^{−}V_{n_{k}}^{m_{n_{k}}} U_{n_{k}}w_{n_{k}}w_{n_{k+1}}^{−}U_{n_{k+1}}^{−}V_{n_{k+1}}W_{n_{k+1}}w_{n_{k+1}} ⋯

Then by a similar argument as before we conclude that the last term is quasi-∼_{w}-reduced and the ∼_{w}-reduced word is obtained by multiplying w_{i}w_{n_{k}}^{−}’s. Let b_{i} = w_{i}w_{n_{k}}^{−} for n_{k−1} ≤ i < n_{k}. Then b_{i}’s appear in the ∼_{w}-reduced word and so {b_{n_{k}} : k < ω} = {b_{i} : i < ω} ∈ c_{j_{0}} and (b_{n_{k}} : k < ω) is a proper sequence. Therefore we can define d_{0} = b_{n_{0}}b_{n_{1}} ⋯ b_{n_{k}}b_{n_{k+1}} ⋯ and d_{k} = b_{n_{k}}b_{n_{k+1}} … generally by Lemma 2.8. Moreover (d_{k} : k < ω) is a proper sequence and {d_{k} : k < ω} belongs to c_{j_{0}} since c_{j_{0}} is properly closed. Let \overline{h}(x) = d_{0}^{−}w_{0}h(x)w_{0}^{−}d_{0}. Then h(δ_{0}) = d_{0}^{−}U_{n_{0}}^{−}V_{n_{0}}U_{n_{0}}d_{0} and h(δ_{k}) = d_{k}^{−}U_{n_{k}}^{−}V_{n_{k}}U_{n_{k}}d_{k} generally, since w_{i_{0}}^{−}d_{0} = w_{n_{k}}^{−}d_{k}. We remark that \overline{h} is also with the empty tail-limit as in Subcase 3.1. Since \overline{h}(δ_{i}) = d_{k}U_{i}^{−}V_{i}U_{i}d_{k}^{−} for n_{k} ≤ i < n_{k+1} and d_{k}^{−}d_{k+1} = d_{k+1}^{−}b_{k}^{−}d_{k+1} ≠ e, we conclude that \overline{h} is standard and Im(h) ⊆ H(C) for some C ∈ ⋓(c_{i} : i ∈ I) by Lemmas 3.16 and 3.17, from which we easily have the conclusion for h. □

Lemma 3.22. Theorem 3.19 holds under the following condition: (1) h(δ_{n}) ≠ e for every n < ω;

(2) for every n < ω V_{n} is a single letter x_{n}, i.e., h(δ_{n}) = W_{n}^{−}x_{n}W_{n};

(3) the tail-limit of W_{n}’s is the empty word;

Proof. (Case 0): W_{n}’s are empty for infinitely many n’s.

(Subcase 0.1) If W_{n}’s are empty for almost all n’s and there exists G_{i^{∗}} such that almost all x_{n}’s belong to G_{i^{∗}}:

The first condition of Theorem 3.19 is satisfied.

(Subcase 0.2) There exist infinitely many n’s for which W_{n}’s are not empty:

Let {i_{n} : n < ω} = {n : W_{n} is empty} and {j_{n} : n < ω} = {n : W_{n} is non-empty} such that i_{n} < i_{n+1} and j_{n} < j_{n+1}. Then h(δ_{i_{n}}δ_{j_{n}}) = x_{i_{n}}W_{j_{n}}^{−}x_{j_{n}}W_{j_{n}}. Let \overline{h}(δ_{2n}) = h(δ_{n}) and \overline{h}(δ_{2n+1}) = h(δ_{i_{n}}δ_{j_{n}}). Then \overline{h} is also with the empty tail-limit. Since \overline{h} satisfies the assumption of Lemma 3.21 we have the conclusion by Lemma 3.17. (Case 1): There exist infinitely many n’s for which W_{n} are non-empty and there exists m > n such that W_{m} is non-empty and W_{n}W_{m}^{−} is ∼_{w}-reduced.

Since W_{n}^{−}x_{n}W_{n}W_{m}^{−}x_{m}W_{m} is cyclically ∼_{w}-reduced, by pairing W_{n}^{−}x_{n}W_{n}W_{m}^{−}x_{m}W_{m} and using Lemma 3.17 we conclude h itself is standard by Case 0 in the proof of Lemma 3.21. (Case 2): Case 0 does not hold and there exists i_{0} such that there’s no maximal element in \overline{W_{i_{0}}} and W_{i_{0}}W_{n}^{−} is not ∼_{w}-reduced for any non-empty W_{n} with n > i_{0}.

Since the tail-limit of W_{n}’s is the empty word, for each n ≥ i_{0} there exists m > n such that W_{n}W_{m}^{−} ≠ e. Let W^{−}V W be the canonical presentation for h(δ_{n}δ_{m}). Then W_{i_{0}}W^{−} is not ∼_{w}-reduced and V is not a single word by Lemma 3.7. Using infinitely many W_{n}^{−}x_{n}W_{n}W_{m}^{−}x_{m}W_{m}’s we can show that h is standard by Lemma 3.17 and Lemma 3.21 (actually Case 2 in the proof of Lemma 3.21). (Case 3) Case 0 does not hold and there exists i_{0} such that there’s a maximal element in \overline{W_{i_{0}}} and W_{i_{0}}W_{i}^{−} is not ∼_{w}-reduced for i > i_{0}. Let W_{i} ≡ U_{i}w_{i} for i ≥ i_{0}. Since the tail-limit of W_{n}’s is empty, for each i ≥ i_{0} there exists i′ > i such that w_{i′} ≠ w_{i}. Hence we have i_{n} < i_{n+1} and j_{n} < j_{n+1} and i^{∗} ∈ I such that w_{i_{n}} ≠ w_{j_{n}} and w_{i_{n}}, w_{j_{n}} ∈ G_{i^{∗}} for all n. Using δ_{i_{n}}δ_{j_{n}}’s we can reduce the conclusion to Lemma 3.21. □

Proof of Theorem 3.19. Let J = {n < ω : h(δ_{n}) = e} and p_{J} : × ×_{n<ω}ℤ_{n} → × ×_{n∈J}ℤ_{n} be the projection. Then we have h(δ_{n}) = h · p_{J}(δ_{n}) for n < ω and hence h = h · p_{J} by Proposition 1.3. Hence we may assume J = ∅. By Lemma 3.12 we have u ∈ × ×^{σ}(G_{i}, c_{i} : i ∈ I) such that the homomorphism \overline{h} : × ×_{n<ω}ℤ_{n} → × ×^{σ}(G_{i}, c_{i} : i ∈ I) defined by: \overline{h}(x) = u^{−}h(x)u is with the empty tail-limit. Then \overline{h} satisfies the conclusion of the theorem by Lemmas 3.21 and 3.22. □

Lemma 3.23. Let G_{i} ∈ S and G = × ×^{σ}(G_{i}, c_{i} : i ∈ I). Suppose that two uncountable H_{0} ∈ H(G_{i}) and H_{1} ∈ H(G_{j}) are compatible for i ≠ j. Then there exist C ∈ c_{i} and D ∈ c_{j} such that H_{0}, H_{1} ⪯ H(C ∪ D).

Proof. We have H ∈ H(G) such that H_{0} ∪ H_{1} ⊆ H. We apply Theorem 3.19 to H. If the case (1) holds, we have H′ ∈ H(G_{k}) such that H ⪯ H′, but this implies that at least one of H_{0} and H_{1} is a subgroup of finitely generated group, which is a contradiction. Hence the case (2) holds and we have the conclusion. □

Corollary 3.24. Let G_{i} ∈ S and G = × ×^{σ}(G_{i}, c_{i} : i ∈ I). For x ∈ X_{G}, one of the following holds:

(1) there exists a unique i ∈ I such that for each H ∈ x there exists H′ ∈ x ∩ H(G_{i}) with H ⪯ H′ and H′ ⪯ H;

(2) for every C ∈ ⋓(c_{i} : i ∈ I), H(C) ∈ x. In case G = ∗(G_{i}, c_{i} : i ∈ I), the statement replaced (2) by the following (2′) holds: (2′) for every C ∈ ⋓^{F}(c_{i} : i ∈ I), H(C) ∈ x.

Proof. Let H_{0} ∈ H(G) be uncountable. By Theorem 3.19 there exists H ∈ H(G) such that H ⪯ H_{0}, H_{0} ⪯ H and

(a) H ∈ H(G_{i}); or

(b) H = H(C) for C ∈ ⋓(c_{i} : i ∈ I). If H_{0} ∈ x, then the above H also belongs to x by the maximality of x.

Suppose that x contains H_{0} ∈ H(G_{i}) such that H_{0} ⪯ H(C) does not hold for any C ∈ ⋓(c_{i} : i ∈ I). Then for each H ∈ x we have H′ ∈ H(G_{i}) such that H ⪯ H′ and hence the case (1) holds.

Otherwise for every H ∈ x we have H ⪯ H(C) for some C ∈ ⋓(c_{i} : i ∈ I). Since {H ∈ H(G) : H ⪯ H(C) for some C ∈ ⋓(c_{i} : i ∈ I)} is a compatible family, the case (2) holds.

By the use of Corollary 3.20 instead of Theorem 3.19 in the preceding proof, we have the conclusion for ∗(G_{i}, c_{i} : i ∈ I). □

4. Application to spaces

Definition 4.1. Let (X_{i}, x_{i}) be pointed spaces. The direct product Π_{i∈I}(X_{i}, x_{i}) is a pointed space (Π_{i∈I}X_{i}, x) where x^{∗}(i) = x_{i} for i ∈ I. The weak join ˜⋁_{i∈I}(X_{i}, x_{i}) is a pointed-subspace of Π_{i∈I}(X_{i}, x_{i}) whose base set is {y ∈ Π_{i∈I}X_{i} : y(i) = x_{i} except for one i}. The join ⋁_{i∈I}(X_{i}, x_{i}) is a pointed space whose base set is the same as that of ˜⋁_{i∈I}(X_{i}, x_{i}) and whose topology is the following:

(1) the restriction of the topology to each X_{i} is the same as that of X_{i};

(2) for arbitrary open neighborhoods U_{i} of x_{i} the union ⋃_{i∈I}{y : y(i) ∈ U_{i}, y(j) = x_{j} for j ≠ i} is a neighborhood of x^{∗}.

First we characterize these two topologies among one point unions. The proof is straightforward and we omit it.

Lemma 4.2. Suppose that the base set of a space X is the disjoint union of X_{i} s with identification of x_{i} s, x^{∗} denotes the identified point, and the topology of X_{i} \ {x_{i}} is the subspace topology of X_{i} for each i.

(1) The topology of X coincides with ˜⋁_{i∈I}(X_{i}, x_{i}), if for S ⊆ X, x^{∗} ∈ \overline{S} is equivalent that x^{∗} ∈ \overline{S ∩ X_{i}} for some i ∈ I or \{i ∈ I : X_{i} ∩ S ≠ ∅} is infinite.

(2) The topology of X coincides with ⋁_{i∈I}(X_{i}, x_{i}), if for S ⊆ X, x^{∗} ∈ \overline{S} is equivalent x^{∗} ∈ \overline{S ∩ X_{i}} for some i ∈ I.

Lemma 4.3. Let G ∈ S and C be an admissible subset of G. Then there exists a unique H_{C} ∈ H_{G} such that H_{C} = Im(h) for some h : × ×_{n<ω}ℤ_{n} → G such that C = {h(δ_{n}) : n < ω}.

Proof. Let C = {h′(δ_{n}) : n < ω} for h′ : × ×_{n<ω}ℤ_{n} → G and let φ : ω → ω be a map such that h(δ_{n}) = h′(δ_{φ(n)}) for each n. For W ∈ W(\{δ_{n}, δ_{n}^{−} : n < ω\}), let \overline{W′} = \overline{W} and W′(α) = δ_{φ(n)}^{±1} when W (α) = δ_{n}^{±1} respectively. Since G ∈ S, there exists only finitely many n’s such that h(δ_{n}) = c for each non-trivial c ∈ G. Hence W′ belongs to W({δ_{n}, δ_{n}^{−} : n < ω}). To see h′(W′) = h(W), let h_{0} : G → S be a homomorphism to an n-slender group S. Then h_{0} · h′(δ_{n}) = h_{0} · h(δ_{n}) = e for almost all n and we get h_{0} · h′(W′) = h_{0} · h(W), which implies the conclusion. □

We remark some relationship between compatibility of subgroups and admissible compatibility of subsets. Let C_{0} and C_{1} be infinite compatible subsets of G and h_{0} and h_{1} be the associated homomorphisms. Then Im(h_{0}) and Im(h_{1}) are compatible and Im(h_{0}) and g^{−} Im(h_{1})g are also compatible for every g ∈ G. When G ≤ × ×_{I}^{σ}ℤ and C_{0} and C_{1} are infinite, C_{0} and g^{−}C_{1}g are never compatible for g ≠ e [5, Theorem 2.3].

A metric space X is mild at x, if for a neighborhood V of x there exists a neighborhood U of x such that U ⊆ V and any null homotopic loop f of its base point x with Im(f) ⊆ U is null homotopic in V.

Lemma 4.4. A one dimensional space is mild. If (X_{i}, x_{i})’s are mild spaces, then spaces Π_{i∈I}X_{i}, ˜⋁_{i∈I}(X_{i}, x_{i}) and ⋁_{i∈I}(X_{i}, x_{i}) are also mild.

The next lemma is straightforward and we omit the proof.

Lemma 4.5. Let f and g be homotopic loops such that Im(f), Im(g) ⊆ U. If every null-homotopic loop whose image is contained in U is nullhomotopic in V, then f and g are homotopic in V.

Lemma 4.6. Let f be a loop in ⋁_{i∈I}(X_{i}, x_{i}). Then, there exists a finite subset F of I such that Im(f) ⊆ ⋁_{i∈F} (X_{i}, x_{i}).

Proof. To show this by contradiction, suppose the negation of the conclusion. Then, we have a_{n} ∈ [0, 1] such that for distinct a_{n} s f(a_{n}) s belong to distinct X_{i}s, i.e. f(a_{n}) ∈ X_{i_{n}} and i_{m} ≠ i_{n} for m ≠ n. We have a neighborhood U_{i_{n}} of x_{i_{n}} for each n such that f(a_{n}) ∉ U_{i_{n}}. There exists an accumulation point a of a_{n} s. Then f(a) = x^{∗}(= x_{i_{n}}), and hence f(a_{n}) ∈ U_{i_{n}} for infinitely many n s, which is a contradiction. □

Lemma 4.7. Let H be an admissibly closed subgroup of G_{i}, U ∈ W^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) be a word weakly equivalent to ⟨⟩. Suppose that

• α_{0} < ⋯ < α_{m} in \overline{U} and \{α_{0}, ⋯, α_{m}\} = \{α ∈ \overline{U} : U(α) ∈

G_{i} \ H};
• U ≡ Y_{0}U(α_{0})Y_{1} ⋯ Y_{m}U(α_{m})Y_{m+1}.

Then there exists Y_{k} (1 ≤ k ≤ m)such that Y_{k} ∼_{w} u for u ∈ H.

Proof. We may assume Y_{k}’s are ∼_{w}-reduced. Then there exist Y_{k} or Y_{k+1} such that the right most letter in Y_{k} belongs to G_{i} or the left most letter in Y_{k+1} belongs to G_{i} by Corollary 2.19, i.e. Y_{k} ≡ Y′_{k}b_{k} or Y_{k+1} ≡ a_{k+1}Y′_{k+1} for a_{k}, b_{k} ∈ G_{i}. Since a_{k}, b_{k} ∈ H if a_{k}, b_{k} exist, a_{k}U(α_{k}), U(α_{k})b_{k}, nor a_{k}U(α_{k})b_{k} does not belong to H. This implies that at least one of Y_{k}’s (1 ≤ k ≤ m) consists of a single letter in H. □

To prove the next theorem we need some notion. A loop f : [a, b] → X is a continuous map with f(a) = f(b). Two paths f : [a, b] → X and g : [c, d] → X are homotopic if there exists a continuous map H whose domain is the quadrangle in the plane with the vertices (a, 0), (b, 0), (c, 1) and (d, 1) such that

H(s, 0) = f(s) for a ≤ s ≤ b,
H(s, 1) = g(s) for c ≤ s ≤ d,
H((1 − t)a + tc, t) = f(a) = g(c) for 0 ≤ t ≤ 1,
H((1 − t)b + td, t) = f(b) = g(d) for 0 ≤ t ≤ 1.

The homotopy class containing a path f is denoted by [f]. The homotopy defined above is usually called “a homotopy relative to end points”. Since we only treat with homotopies relative to end points, we call “homotopy” instead of “homotopy relative to end points” for short. Let L(X, x) be the set of loops f : [a, b] → X with f(a) = f(b) = x and [f] ∈ π_{1}(X, x) be the homotopy class containing f ∈ L(X, x).

Let X = ˜⋁_{i∈I}(X_{i}, x_{i}) or ⋁_{i∈I}(X_{i}, x_{i}) and f ∈ L(X, x). Then {s : f(s) ≠ x} is an open subset of [0, 1] and is expressed as the sum of pair-wise disjoint open intervals, i.e.,

• {s : f(s) ≠ x} = ⋃_{α∈M}(a_{α}, b_{α});

• M is linearly ordered and b_{α} < a_{β} if α < β.

We define W^{f} by: \overline{W^{f}} = M and W^{f}(α) = [f ↾ [a_{α}, b_{α}]] ∈ π_{1}(X_{i}, x_{i}) where f([a_{α}, b_{α}]) ⊆ X_{i}.

Lemma 4.8. If [f] = e, then [W^{f}]_{w} = e.

Proof. Suppose that [W^{f}]_{w} ≠ e. Then we have homomorphisms h_{i} : π_{1}(X_{i}, x_{i}) → S_{i} (i ∈ F) to n-slender groups S_{i} such that h((W^{f})_{F}^{h}) ≠ e. Then M_{0} = {α ∈ W^{f} : h(W^{f}(α)) ≠ e} is finite and

h((W^{f})_{F}^{h}) = ⋯ h(W^{f}(α)) ⋯ h(W^{f}(β)) ⋯

for α < β and α, β ∈ M_{0}. We remark that the right hand side of the equation is a word for an element of ∗_{i∈F} S_{i}.

Let H : [0, 1] × [0, 1] → X be a homotopy from f to the constant, i.e. H(s, 0) = f(s), H(s, 1) = H(0, t) = H(1, t) = x. Let ⋃_{n∈K} O_{n} = {(s, t) : H(s, t) ≠ x} where each O_{n} is connected and O_{m} ∩ O_{n} = ∅ for m ≠ n and J be the set of n’s such that O_{n} contains (a_{α}, b_{α}) × {0} satisfying h(W^{f}(α)) ≠ e. Then J is finite. We write O_{m} ≪ O_{n}, when O_{m} and [0, 1] × {0} are contained in distict connected components of [0, 1] × [0, 1] \ O_{n}.

Let f_{n}(s) = f(s) if (s, 0) ∈ O_{n} and f_{n}(s) = x otherwise and H_{n}(s, t) = H(s, t) if (s, t) ∈ O_{n} and H_{n}(s, t) = x otherwise. Then H_{n} is a homotopy from f_{n} to the constant and consequently [f_{n}] = e. Since [f_{n}] ∈ π_{1}(X_{i}, x_{i}) for some i, (W^{f_{n}})_{F}^{h} = e. A finite set {O_{n} : n ∈ J} is partially ordered by ≪ and so we have a reduction of the word ⋯ h(W^{f}(α)) ⋯ h(W^{f}(β)) ⋯ to the empty word according to ≪, which contradicts h((W^{f})_{F}^{h}) ≠ e. □

For x ∈ X, let c_{x} = {f_{∗}({[e_{n}] : n < ω}) ⊆ π_{1}(X, x) | f : ℍ → X, f(o) = x}, where [e_{n}] is the element corresponding to the winding to the n-th circle in ℍ.

Lemma 4.9. Suppose that π_{1}(X, x) ∈ S. Then the family c_{x} is a properly closed compatible family, i.e. c_{x} ∈ C_{π_{1}(X,x)}.

Proof. For given continuous map f_{0}, f_{1} from (ℍ, o) to (X, x), define f : (ℍ, o) → (X, x) so that f ·e_{2n} = f_{0} ·e_{n} and f ·e_{2n+1} = f_{1} ·e_{n}. Then, we have {f_{∗}(δ_{n}) : n < ω} = {f_{0∗}(δ_{n}) : n < ω} ∪ {f_{1∗}(δ_{n}) : n < ω} and hence c_{x} is compatible.

Let {h(δ_{n}) : n < ω} ∈ c_{x} for h : × ×_{n<ω}ℤ_{n} → G. Then we have a continuous map f : (ℍ, o) → (X, x) such that {h(δ_{n}) : n < ω} ⊆ {f_{∗}(δ_{n}) : n < ω}. There exists a map φ : ω → ω such that h(δ_{n}) = f_{∗}(δ_{φ(n)}) and {n < ω|φ(n) = m} is finite for each m < ω. Modifying f we have f_{2} : (ℍ, o) → (X, x) such that f_{2∗}(δ_{n}) = h(δ_{n}). By Proposition 1.3 we have h = f_{2∗}. For a proper sequence (y_{n} : n < ω) in × ×_{n<ω}ℤ_{n} we have a continuous map ψ : (ℍ, o) → (ℍ, o) such that ψ_{∗}(δ_{n}) = y_{n} for n < ω. Finally we have {h(y_{n}) : n < ω} = {(f ∘ ψ)_{∗}(δ_{n}) : n < ω} ∈ c_{x}. □

Theorem 4.10. Let X_{i} be path-connected metric spaces such that π_{1}(X_{i}, x_{i}) ∈ S and X_{i} are mild at x_{i}. Then π_{1}( ˜⋁_{i∈I}(X_{i}, x_{i}), x) is canonically isomorphic to × ×^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) and π_{1}(⋁_{i∈I}(X_{i}, x_{i}), x) is canonically isomorphic to ∗(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I).

Proof. By Lemma 4.8 we define φ : π_{1}(X, x) → × ×^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) by: φ([f]) = [W^{f}]_{w}. The fact that range of φ is in × ×^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) follows from W^{f} ∈ W^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I).

To see the surjectivity, it suffices to show the existence of a loop f with W^{f} ≡ U for a given U ∈ W^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I). There exist C_{i} ∈ c_{x_{i}} such that U ∈ W^{σ}(⋃_{i∈I} C_{i}). We have loops f_{u} for u ∈ C_{i} such that [f_{u}] = u and almost all Im(f_{u})’s are contained in an arbitrary neighborhood of x_{i}. Patching f_{u}’s for u ∈ ⋃_{i∈I} C_{i} to the intervals of the complement of the Cantor ternary set, we have a loop f such that W^{f} ≡ U. The continuity of f follows from the topology of ˜⋁_{i∈I}(X_{i}, x_{i}).

The proof of the injectivity of φ is a modification of the proof of [3, Theorem A.1]. Let f : [0, 1] → X = ⋁_{i∈I}(X_{i}, x_{i}) be a loop with the base point x and W^{f} ∼_{w} e. We show that f is null-homotopic in the following. Since the image of a loop is separable, we may assume that an index set I is countable. We choose a map η : ω → I so that {m : η(m) = i} is infinite for each i and define ξ : ω → ω by:

ξ(m) = \begin{cases}max\{k : η(k) = η(m), k < m\}, if such a k exists; \\ 0, otherwise. \end{cases}

We note that ξ is a strictly increasing function. Choose a neighborhood base {O_{in} : n < ω} of each x_{i} in X_{i} so that a null-homotopic loop with base point x_{i} whose image is contained in O_{i n+1} is null-homotopic in O_{in}. For the notational convenience we assume O_{i0} = O_{i1} = X_{i}. Define a subgroup H_{in} of π_{1}(X_{i}, x_{i}) by: H_{in} = {[f] | f : [0, 1] → X_{i}, f(0) = f(1) = x, Im(f) ⊆ O_{in}}.

Let f^{−1}(X \ {x}) = ⋃_{n<μ}(a_{n}, b_{n}) where (a_{m}, b_{m}) ∩ (a_{n}, b_{n}) = ∅ for m ≠ n and μ ≤ ω.

We define parts of a homotopy H : [0, 1] × [0, 1] → X between f and the constant map at x and also auxiliary notions by induction. To describe our inductive definition, we introduce some notion.

Let Seq be the set of finite sequences of ordered pairs of nonnegative integers and closed intervals [u, v] where u, v ∈ {a_{n}, b_{n} : n ≤ μ}. We shall define Σ as a subset of Seq by induction along a construction of a homotopy. For σ ∈ Seq, the length of σ is denoted by |σ| and σ = ⟨σ(0), ⋯, σ(n−1)⟩, where n = |σ| and each σ(i) is an ordered pair. The sequence obtained by adding (i, k) to σ is denoted by σ ∗ ⟨(i, k)⟩ and the empty sequence is denoted by ⟨ ⟩. Let R_{⟨ ⟩} = [0, 1] × [0, 1] and ⟨ ⟩ ∈ Σ. Let H(s, 1) = f(s) and H(s, 0) = H(0, t) = H(1, t) = x for 0 ≤ s, t ≤ 1 and also f_{⟨ ⟩} = f. Our construction goes on the length of a sequence σ and the first stage is on the empty sequence ⟨ ⟩. Since the first stage is a very special case of a general stage σ, we only state a general stage.

(Stage σ ) Let i = η(|σ|) and n = |σ| + 1. In this stage we concern with O_{in}. We note Im(f_{σ})∩(X_{i}\O_{i ξ(|σ|)}) = ∅ according to our construction and consequently W^{f_{σ}}(α) ∈ G_{i}\H_{i |σ|}) = ∅ for every α ∈ \overline{W^{f_{σ}}}. We are given f_{σ} and R_{σ} = [a_{σ}, b_{σ}]×[c_{σ}, d_{σ}] so that W^{f_{σ}} ∼_{w} ⟨ ⟩, and H(s, t) has been defined on ∂R_{σ} so that H(s, d_{σ}) = f_{σ}(s) for a_{σ} ≤ s ≤ b_{σ} and H(a_{σ}, t) = H(b_{σ}, t) = H(s, c_{σ}) = x for a_{σ} ≤ s ≤ b_{σ}, c_{σ} ≤ t ≤ d_{σ}. Our construction will be performed on finitely many sub-stages by induction. Since the 0-th sub-stage is a special case of the p-th stage, we only state about the p-th stage. (Sub-stage p) Let g_{p}(s) = H(s, c_{σ} + (1/2+1/2^{p+1})(d_{σ} − c_{σ}))) for s ∈ [a_{σ}, b_{σ}].

We have a_{σ} = a_{p0} ≤ a_{p1} < a_{p2} < ⋯ < a_{p 2m_{p}} ≤ a_{p 2m_{p}+1} = b_{σ} such that:

• g_{p}(a_{pj}) = x for 0 ≤ j ≤ 2m_{p} + 1;
• g_{p}([a_{p 2k−1}, a_{p 2k}]) ⊆ X_{i} for 1 ≤ k ≤ m_{p};
• g_{p}([a_{p 2k}, a_{p 2k+1}]) ∩ (X_{i} \ O_{in}) = ∅ for 0 ≤ k ≤ m_{p}.

We remark that m_{p} may be 0. (Case 1) [g_{p} ↾ [a_{p 2k−1}, a_{p 2k}]] ∈ H_{in} for every 1 ≤ k ≤ m_{p}:

For each k we have a homotopy

G_{k} : [a_{p 2k−1}, a_{p 2k}]×[(c_{σ} +d_{σ})/2, c_{σ} +(1/2+1/2^{p+1})(d_{σ} −c_{σ})] → O_{i ξ(n−1)}

so that G_{k}(s, d_{σ}) = g_{p}(s) and G_{k}(s, (c_{σ}+d_{σ})/2) ∈ O_{in} for s ∈ [a_{p 2k−1}, a_{p 2k}] by Lemma 4.5. Define

H(s, t) = g_{p}(s) for s ∈ [a_{σ}, b_{σ}] \ ⋃_{k≤m_{p}}[a_{p 2k}, a_{p 2k+1}], t ∈ [(c_{σ} + d_{σ})/2, c_{σ} + (1/2+1/2^{p+1})(d_{σ} − c_{σ})] and H(s, t) = G_{k}(s, t) for s ∈ [a_{p 2k−1}, a_{p 2k}], t ∈ [(c_{σ}d_{σ})/2, c_{σ} + (1/2+1/2^{p+1})(d_{σ} − c_{σ})].

Let σ ∗ ⟨(p, [a_{σ}, b_{σ}])⟩ ∈ Σ and f_{σ∗⟨(p,[a_{σ},b_{σ}])⟩}(s) = H(s, (c_{σ} + d_{σ})/2) for s ∈ [a_{σ}, b_{σ}] and R_{σ∗⟨(p,[a_{σ},b_{σ}])⟩} = [a_{σ}, b_{σ}] × [c_{σ}, (c_{σ} + d_{σ})/2]. We finish this stage. (Case 2) Otherwise, i.e. [g_{p} ↾ [a_{p 2k−1}, a_{p 2k}]] ∈ G_{i} \ H_{in} for some 1 ≤ k ≤ m_{p}:

We have a_{σ} = b_{p0} ≤ b_{p1} < b_{p2} < ⋯ < b_{p 2n_{p}} ≤ b_{p 2n_{p}+1} = b_{σ} such that:

• n_{p} ≥ 1;
• g_{p}(b_{pj}) = x for 0 ≤ j ≤ 2n_{p} + 1;

• g_{p}([b_{p 2k−1}, b_{p 2k}]) ⊆ X_{i} and [g_{p} ↾ [b_{p 2k−1}, b_{p 2k}]] ∉ H_{in} for 1 ≤

k ≤ m_{p};
• g_{p}([a_{p 2k}, a_{p 2k+1}]) ∩ (X_{i} \ O_{in}) = ∅ for 0 ≤ k ≤ m_{p}.

By Lemma 4.7 [g_{p} ↾ [b_{p 2k}, b_{p 2k+1}]] ∈ H_{in} for some 1 ≤ k ≤ n_{p}−1. Let K = {k : 1 ≤ k ≤ n_{p}−1, [g_{p} ↾ [b_{p 2k}, b_{p 2k+1}]] ∈ H_{in}}.

For k ∈ K, by Corollary 2.18 we have b_{p 2k} ≤ u_{kα} < v_{kα} ≤ b_{p 2k+1} such that:

• g_{p}(u_{kα}) = g_{p}(v_{kα}) = x and (u_{kα}, v_{kα}) ∩ (u_{kβ}, v_{kβ}) = ∅ for α ≠ β;

• W^{g_{p}↾[u_{kα},v_{kβ}]} ∼_{w} ⟨⟩;
• g_{p}(s) ∈ X_{i} for s ∈ [a_{p 2k}, a_{p 2k+1}] \ ⋃_{α}(u_{α}, v_{α}).

Let σ∗⟨(p, [u_{α}, v_{α}])⟩ ∈ Σ, f_{σ∗⟨(p,[u_{α},v_{α}])⟩} = g_{p} ↾ [u_{α}, v_{α}] and R_{σ∗⟨(p,[u_{α},v_{α}])⟩} = [u_{α}, v_{α}] × [c_{σ} + (1/2 + 1/2^{p+2})(d_{σ} − c_{σ}), c_{σ} + (1/2 + 1/2^{p+1})(d_{σ} − c_{σ})].

Define H(s, t) = H(s, c_{σ} + (1/2 + 1/2^{p+1})(d_{σ} − c_{σ}))

for s ∈ [a_{σ}, b_{σ}] \ ⋃_{α}(u_{α}, v_{α}),

t ∈ [c_{σ} + (1/2+1/2^{p+2})(d_{σ} − c_{σ}), c_{σ} + (1/2+1/2^{p+1})(d_{σ} − c_{σ})] and H(s, t) = x for s ∈ ⋃_{α}(u_{α}, v_{α}), t = c_{σ} + (1/2+1/2^{p+2})(d_{σ} − c_{σ}). Now H(s, c_{σ} + (1/2+1/2^{p+2})(d_{σ} − c_{σ})) has been defined for every

s ∈ [a_{σ}, b_{σ}] and Case 2 has been completed.

Since m_{p+1} < n_{p} ≤ m_{p}, the sub-stages stop in a finite step. Moreover n_{p} ≥ 2 when n_{p} ≥ 1 by Lemma 4.7 and hence the sub-stages stop in Case 1.

Now we have described our inductive construction. After all σ-stages finish, there may be points (s, t) on which H(s, t)’s have not been defined. We define H(s, t) = x for such points (s, t). What is remaining to show is the continuity of the map H. At each (s, t) for which we defined H(s, t) = x just now, there exist σ_{n} ∈ Σ such that |σ_{n}| = n and (s, t) ∈ int(R_{σ_{n}}). Since H(R_{σ_{n}}) converge to x when n → ∞, H is continuous at such points (s, t). If H(s, t) ≠ x, H is defined on a neighborhood of (s, t) by one or two homotopies and the continuity of H at (s, t) follows from those of homotopies. The remaining case is when: H(s, t) = x and H(s, t) is defined in some stage σ. Choose a neighborhood U = ⋂_{k=0}^{q} O_{i_{k}n_{k}} of x. Let D_{τ} be the union of [a_{τ}, b_{τ}] × {d_{τ}} and the set of (s, t) ∈ R_{τ} for which H(s, t) is defined in Stage τ. There exists l_{0} such that f_{σ} ⊆ ⋂_{k=0}^{q} O_{i_{k} n_{k}+1} for |σ| ≥ l_{0}. This implies H(R_{σ}) ⊆ U. For each σ there are only finitely many extensions τ of σ with |τ| = |σ| + 1 such that Im(f_{τ}) are not contained in ⋂_{k=0}^{q} O_{i_{k} n_{k}+1}. Consequently there are only finitely many stages σ’s such that H(s, t) ∉ U for some (s, t) ∈ D_{σ}. Since H is continuous on each D_{σ}, there exists a neighborhood of x on which H takes values in U. Now we have shown the continuity. □

Remark 4.11. (1) In the preceding proof we auxiliarily use subgroups H_{O} of π_{1}(X, x) for a neighborhood O of x. For spaces in a certain class, c_{x} is closely related to the class {H_{O} : O is a neighborhood of x} as we see in the next section. Here we explain why we used subgroups H_{in} and neighborhoods O_{in} in the preceding proof. Suppose that f : [0, 1] → X is a loop satisfying:

(a) f^{−1}(X_{i} \ {x_{i}}) = ⋃_{n<ω}(a_{n}, b_{n}) and (a_{m}, b_{m}) ∩ (a_{n}, b_{n}) = ∅ for m ≠ n;

(b) each f ↾ [a_{n}, b_{n}] is null-homotopic.

Let y ∈ X_{i} be a point which is distinct from x_{i} but path-connected to x_{i}. For each n we have a homotopy G_{n} from f ↾ [a_{n}, b_{n}] to the constant map whose range contains y. We cannot patch these homotopies G_{n} to one homotopy in general. Thus we controlled the sizes of homotopies so that the constructed map becomes continuous.

(2) Here we consider Theorem 4.10 in case the index set I = {0, 1}, i.e. (X_{0}, x_{0})∨(X_{1}, x_{1}). Then we have π_{1}((X_{0}, x_{0})∨(X_{1}, x_{1})) = (π_{(}X_{0}, x_{0}), c_{x_{0}})∗ (π_{(}X_{1}, x_{1}), c_{x_{1}}). If X_{0} has a good local property at x_{0} or X_{1} has a locally good property at x_{1}, at least one of c_{x_{0}} and c_{x_{1}} consists of only finite sets and hence π_{1}((X_{0}, x_{0}) ∨ (X_{1}, x_{1})) is isomorphic to the usual free product π_{1}sy(X_{0}, x_{0}) ∗ π_{1}(X_{1}, x_{1}) [7, 4].

(3) There are natural topologies of one point unions other than ⋁_{i∈I}(X_{i}, x_{i}) or ˜⋁_{i∈I}(X_{i}, x_{i}). Here we mention two kinds of them. Suppose that X_{i} is a metric space d_{i} for i ∈ I. Define d(x^{∗}, x) = d_{i}(x^{∗}, x) for x ∈ X_{i} and let {x : d(x^{∗}, x) < ε} be a neighborhood base of x^{∗} for ε > 0. This defines a different topology. For another one, let F be a filter on I and define O to be open if O ∩ X_{i} is open for every i and {i ∈ I : X_{i} ⊆ O} ∈ F.

5. Algebraic cut point

Definition 5.1. A point x ∈ X_{G} is an algebraic cut point, if there exist subgroups G_{0}, G_{1} of G, and maximal elements c_{0} ∈ C_{G_{0}} and c_{1} ∈ C_{G_{1}} such that G = (G_{0}, c_{0}) ∗ (G_{1}, c_{1}), x ∩ H(G_{0}) ∈ X_{G_{0}}, x ∩ H(G_{1}) ∈ X_{G_{1}}, and x is the unique point satisfying H(C) ∈ x for every C ∈ c_{0} ∪ c_{1}.

Lemma 5.2. Suppose that x ∈ X_{G} contains a conjugate of a subgroup H ∈ H(G). Then x also contains H.

Proof. Let F ⋐ x and u^{−1}Hu ∈ x. Then, by the compatibility of x, there exists H_{0} ∈ H(G) such that ⋃ F ∪u^{−1}Hu ⊆ H_{0}. Since ⋃ F ∪H ⊆ ⟨H_{0} ∪ {u}⟩ ∈ H(G), H ∈ x by the maximality of x. □

Theorem 5.3. Let G ∈ S and x ∈ X_{G} be an algebraically cut point and G_{0} and G_{1} be the associated subgroups. Then X_{G} is homeomorphic to the one point union of X_{G_{0}} and X_{G_{1}} under the identification of x∩H_{G_{0}} and x ∩ H_{G_{1}}.

Proof. Let x_{0} = x ∩ H_{G_{0}}, x_{1} = x ∩ H_{G_{1}}, and x^{∗} be the identified point of (X_{G_{0}}, x_{0}) ∨ (X_{G_{1}}, x_{1}). Define f : X_{G} → (X_{G_{0}}, x_{0}) ∨ (X_{G_{1}}, x_{1}) by:

f(y) = \{ \begin{matrix}y ∩ H(G_{0}) & if y ≠ x and y ∩ H(G_{0}) ∈ X_{G_{0}}, \\ y ∩ H(G_{1}) & if y ≠ x and y ∩ H(G_{1}) ∈ X_{G_{0}}, \\ x^{∗}, & if y = x. \end{matrix}

If H ∈ H(G) does not belong to x, H is uncountable and contains a uncoutable group in H(G_{0}) or H(G_{1}) by Theorem 3.19 (or Corollary 3.20) and Lemma 5.2.

Hence it is easy to see that f is well-defined bijection. Now it suffices to show the continuity and inverse continuity of f. Since the inverse continuity is straightforward, we only prove the continuity. Suppose that lim_{n→∞} y_{n} = y. Without loss of generality we may assume y_{n} ≠ x. Let H_{n} ∈ f(y_{n}). We want to choose H′_{n} ∈ f(y_{n}) which assure lim_{n→∞} f(y_{n}) = f(y). If H_{n} is finitely generated, we can choose H′_{n} as the trivial group. Hence we suppose that H_{n} is uncountable. We have H′′_{n} ∈ y_{n} (n < ω) which assure lim_{n→∞} y_{n} = y. Here H_{n} ⪯ H′′_{n}. By the above fact due to Theorem 3.19, H′′_{n} contains an uncountable H′_{n} in H(G_{0}) or H(G_{1}) such that H′′_{n} ⪯ H′_{n}. Now H′_{n} ∈ f(y_{n}) and H′_{n} s assure lim_{n→∞} f(y_{n}) = f(y). □

Theorem 5.4. Let G_{i} ∈ S for i ∈ I and G = × ×^{σ}(G_{i}, c_{i} : i ∈ I). Suppose that x_{i} = {H ≤ G : H ≤ H(C) for some C ∈ c_{i}} ∈ X_{G_{i}} for each i ∈ I. Then X_{G} = ˜⋁_{i∈I}(X_{G_{i}}, x_{i}).

Proof. Let x^{∗} ∈ X_{G} be the element containing all H(C) for countable C ∈ ⋓(c_{i} : i ∈ I) s. Then x^{∗} ∩ H_{G_{i}} = x_{i} ∈ X_{G_{i}} for every i ∈ I by the assumption. Let Y_{i} = {y ∈ X_{G} : y ∩ H_{G_{i}} ∈ X_{G_{i}}}. Then X_{G} = ⋃_{i∈I} Y_{i}, {x^{∗}} = Y_{i} ∩ Y_{j} for i ≠ j, and Y_{i} is homeomorphic to X_{G_{i}} as in the proof of Theorem 5.3. To apply Lemma 4.2, let S ⊆ X \ \{x^{∗}\}. If i ∈ I : S ∩ Y_{i} ≠ ∅ is finite, x ∈ \overline{S} is equivalent to x ∈ \overline{S ∩ X_{i}} for some i ∈ I. On the other hand, if i ∈ I : S ∩ Y_{i} ≠ ∅ is infinite, there exist x_{n} ∈ S ∩ Y_{i_{n}} such that i_{m} ≠ i_{n} for m ≠ n. For H_{n} ∈ x_{n}, there exists H′_{n} ∈ x_{n} such that H_{n} ⪯ H′_{n}, H′_{n} ∈ x_{n} and H′_{n} ≤ G_{i_{n}}. Then {a_{n} : n < ω} is addmissible for any a_{n} ∈ H′_{n} and H({a_{n} : n < ω}) ∈ x^{∗}, which implies lim_{n→∞} x_{n} = x^{∗}. Now the conclusion follows from Lemma 4.2. □

Theorem 5.5. Let G_{i} ∈ S for i ∈ I and G = ∗(G_{i}, c_{i} : i ∈ I). Suppose that x_{i} = {H ≤ G : H ≤ H(C) for some C ∈ c_{i}} ∈ X_{G_{i}} for each i ∈ I. Then X_{G} = ⋁_{i∈I}(X_{G_{i}}, x_{i}).

Proof. Let x^{∗} ∈ X_{G} be the element containing all H(C) for countable C ∈ ⋓^{F}(c_{i} : i ∈ I). Then x^{∗} ∩ H_{G_{i}} = x_{i} ∈ X_{G_{i}} for every i ∈ I by the assumption as in the proof of Theorem 5.4. If lim_{n→∞} x_{n} = x^{∗} for x_{n} ∈ X_{G} \ {x^{∗}}, we have uncountable H_{n} ∈ x_{n} (n < ω) such that H({a_{n} : n < ω}) ∈ x^{∗} for every a_{n} ∈ H_{n}. Since x_{n} ∈ Y_{i} for some i ∈ I, we may assume that each H_{n} is a subgroup of G_{i} for some i ∈ I by Corollary 3.20 and Lemma 5.2. G does not contain an element expressed by words using letters from infinitely many G_{i} s. Hence, there exists a finite subset F of I such that H_{n} ⊆ ⋃_{i∈F} G_{i}. On the other hand, we have \overline{S ∩ Y_{i}} ⊆ Y_{i}. Now we have that x^{∗} ∈ \overline{S} if and only if x^{∗} ∈ \overline{S ∩ Y_{i}} for some i ∈ I. The conclusion follows from Lemma 4.2. □

For a point x ∈ X, let p_{x} = {H ≤ π_{1}(X, x) | H ≤ f_{∗}(π_{1}(ℍ, o)), f : ℍ → X, f(o) = x}. In case one of X has a good local property, c_{x}, defined just before Lemma 4.9, is the set of finite subsets of π_{1}(X, x) and p_{x} contains only finitely generated subgroups of π_{1}(X, x). We remark that c_{x} and p_{x} are not defined for an abstract group π_{1}(X, x) but for the fundamental group as a functor from the category of pointed spaces.

Proposition 5.6. For a locally path-conneted, path-connected, one dimensional wild metric space X, x is a cut point X if and only if the maximal compatible subfamily p_{x} of H_{π_{1}(X,x)} containing all H(C) for C ∈ c_{x} is an algebraic cut point in π_{1}(X, x).

Proof. Let G = π_{1}(X), then by [1, Theorem 5.1] X_{G} is homeomorphic to X and the maximal compatible subfamily of H_{π_{1}(X)} containing all H(C) for C ∈ c_{x} is p_{x}. Then, the homeomorphism f : X_{G} → X is given by f(p_{x}) = x.

If x is a cut point of X, we have X_{0} and X_{1} so that X is a one-point union of X_{0} and X_{1} and x is the common point of X_{0} and X_{1}. Let p_{x}^{0} and p_{x}^{1} be the maximal compatible families related to x in π_{1}(X_{0}) and π_{1}(X_{1}) respectively. Since G = (π_{1}(X_{0}), c_{x}^{0})∗(π_{1}(X_{1}), c_{x}^{1}) by Theorem 4.10 and π_{1}(X_{0}) and π_{1}(X_{1}) are admissibly closed by Theorem 3.19, it is easily seen that p_{x} is a cut point of G. On the other hand, if p_{x} is an algebraically cut point, then X_{G} is a one-point union of X_{G_{0}} and X_{G_{1}} by Theorem 5.4 with the common point p_{x}. Hence, x = f(p_{x}) is a cut point of X. □

In the above propositions, one dimensionality, mildness more precisely, is essential as the well-known example shows, i.e. the one point union of cones Cℍ ∨ Cℍ.

6. A class of spaces

In this section we define a class of spaces to which we can apply our results in the preceding section.

Definition 6.1. A class of spaces T is defined inductively as follows:

(1) Each locally path-connected, path-connected one-dimensional metric space which is not semi-locally simply-connected at any point belongs to the class T ;

(2) if spaces X_{n} belong T , the direct product Π_{n<ω}X_{n} belongs to T ;

(3) if spaces X_{i} belong T , the join ⋁_{i∈I}(X_{i}, x_{i}) and the weak join ˜⋁_{i∈I}(X_{i}, x_{i}) belong to T for arbitrary x_{i} ∈ X_{i}.

Lemma 6.2. Let X be a space in T .

(a) X is wild at any point;

(b) X is mild at any point;

(c) A subset S of X is closed if and only if S is closed under convergent sequece, i.e. lim_{n→∞} x_{n} = x and x_{n} ∈ S (n < ω) imply x ∈ S;

(d) for each homomorphism h : π_{1}(ℍ, o) → π_{1}(X, x) there exist a point x^{∗} ∈ X, a path g from x^{∗} to x and a spatial homomorphism \overline{h} : π_{1}(ℍ, o) → π_{1}(X, x^{∗}) such that h = φ_{g} · \overline{h}.

Proof. The proof is an induction on the construction of spaces in T . The initial case follows from [6, Theorem 1.1] and Lemma 4.4. Since proofs for (a), (b) and (c) are straightforward, we only prove (d). Case (2) ： Suppose that spaces X_{n} ∈ T satisfisfy (d).

Let h : π_{1}(ℍ, o) → π_{1}(Π_{n<ω}X_{n}, x^{∗}) be a homomorphism for x^{∗} ∈ Π_{n<ω}X_{n}. Since direct products commute with taking the fundamental groups, we have π_{1}(Π_{n<ω}X_{n}, x^{∗}) = Π_{n<ω}π_{1}(X_{n}, x^{∗}(n)). Let p_{m} :

Π_{n<ω}X_{n} → X_{m} be the projection. By the induction hypothesis, for each n we have a path g_{n} from x^{∗}(n) to x(n) ∈ X_{n} and a spatial homomorphism \overline{h}_{n} : π_{1}(ℍ, o) → π_{1}(X_{n}, x_{n}^{∗}) such that q_{n} ∘ h = φ_{g_{n}} ∘ \overline{h}_{n}. Then p_{n} ∘ g = g_{n} defines a path from x^{∗} in Π_{n<ω}X_{n} and (p_{n})_{∗} ∘ \overline{h} = \overline{h}_{n} defines a spatial homomorphism and we have h = φ_{g} ·\overline{h} and consequently Π_{n<ω}X_{n} satisfies (d). Case (3) ： Suppose that spaces X_{i} ∈ T satisfisfy (d).

Since X_{i} s satisfy the condition of Theorem 5.4 and 5.5, we have π_{1}( ˜⋁_{i∈I}(X_{i}, x_{i})) ≃ × ×^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) and π_{1}(⋁_{i∈I}(X_{i}, x_{i}), x) ≃ ∗(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I). By Theorem 3.19 and Corollary 3.20, ˜⋁_{i∈I}(X_{i}, x_{i}) and ⋁_{i∈I}(X_{i}, x_{i}) satisfy (c). □

Lemma 6.3. [1, Lemma 3.5] Let G_{m} (m < ω) be groups in S, G = Π_{m<ω}G_{m}, and p_{m} : G → G_{m} the projection. Define p_{m}^{∗}(x) = {p_{m}(H) : H ∈ x} for x ∈ X_{G}. If X_{G_{m}} is non-empty, then p_{m}^{∗}(x) belongs to X_{G_{m}}.

Lemma 6.4. [1, Theorem 3.6] Let G_{m} (m < ω) be groups in S, G = Π_{m<ω}G_{m} and I = {m < ω : X_{G_{m}} ≠ ∅}. Then the following hold:

(1) X_{G} ≃ Π_{m∈I}X_{G_{m}}, if I is non-empty;

(2) X_{G} is a one-point space, if I is empty and the set {m < ω : G_{m} ≠ {e}} is infinite.

(3) X_{G} is empty, otherwise.

Lemma 6.5. Let G = π_{1}(X) for a space X ∈ T .

(1) Let c ∈ C_{G} be maximal and containing an infinite set. Then there exists a unique x ∈ X_{G}(≅ X) such that x = {H ≤ G : H ≤ H(C), C ∈ c}.

(2) Let x ∈ X_{G}. Then there exists c ∈ C_{G} such that x = {H ≤ G : H ≤ H(C), C ∈ c}. This c is a maximal compatible family and containing an infinite set.

To state a statement exactly we recall base-point-change isomorphisms on fundamental groups. For a path g from x to y, φ_{g} denotes the base-point-change isomorphism from π_{1}(X, x) to π_{1}(X, y) [8, 1.8.3].

Theorem 6.6. Let X be a space in the class T . Then, for each x ∈ X there exists a unique u ∈ X_{π_{1}(X,x)} such that H(C) ∈ u for every C ∈ c_{x}. For x_{0} ∈ X, f : X → X_{π_{1}(X,x_{0})} defined by f(x) = {φ_{g}(H) : H ∈ u} is a homeomorphism bitween X and X_{π_{1}(X,x)}, where g is a path from x to x_{0} and in particular lim_{n→∞} y_{n} = y is equivalent to lim_{n→∞} f(y_{n}) = f(y). Moreover, f(x) does not depend on the choice of a path g.

Proof. We prove this by induction on the construction of the class T . In the initial case (1) the conclusion is given by [1, Theorem 5.1], which is essentially reduced to [6, Theorem 1.1, Theorem 1.4]. The fact that the choice of a path g does not effect f(x), because H(φ_{g}(C)) is conjugate to H(φ_{g′}(C)) for two paths g, g′ from x to x_{0} and C ∈ c_{x} and hence H(φ_{g}(C)) ⪯ H(φ_{g′}(C)) and H(φ_{g}(C)) ≺ H(φ_{g′}(C)). This reasoning is also valid for the induction step, we do not mention the choice of a path anymore.

Now for the induction step (2), suppose that for X_{n} we have a homeomorphism f_{n} : X_{n} → X_{π_{1}(X_{n},p_{(}x^{∗}))} satisfying the induction hypothesis. For x ∈ X = Π_{n<ω}X_{n}, define f(x) = f_{n}(p_{n}(x)), where p_{n} : Π_{n<ω}X_{n} → X_{n} is the projection for each n < ω. We indentify X_{π_{1}(X,x^{∗})} with Π_{n<ω}X_{π_{1}(X_{n},p_{n}(x^{∗}))} by Lemmas 6.3 and 6.4. Since f_{n} s are homeomorphism, f is a homeomorphism. For x ∈ X, we have c_{x} = {C : countable C ⊆ Π_{n<ω}C_{n}, C_{n} ∈ c_{p_{n}(x)}}. Since c_{p_{n}(x)} determines a unique point in X_{π_{1}(X_{n},p_{n}(x))}, c_{x} determines a unique point in X_{π_{1}(X,x)} by Lemma 6.4.

For the induction step (3), suppose that for X_{i} we have a homeomorphism f_{i} : X_{i} → X_{π_{1}(X_{i},x_{i})} satisfying the induction hypothesis for each i ∈ I. Let x^{∗} be the identified point. Then, π_{1}( ˜⋁_{i∈I}(X_{i}, x_{i}), x^{∗}) is canonically isomorphic to× ×^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) and π_{1}(⋁_{i∈I}(X_{i}, x_{i}), x^{∗}) is canonically isomorphic to ∗(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I) by Theorem 4.10. Let u_{i} be the point in X_{π_{1}(X_{i},x_{i})} containing all H(C) for C ∈ c_{x_{i}}, i.e. f_{1}(x_{i}) = u_{i}. Let x ∈ X. If x ∈ X_{i} \ {x_{i}}, f(x) is determined by c_{x} which is in X_{π_{1}(X,x)}. On the other hand the difference bitween c_{x} in X_{π_{1}(X,x)} and c_{x} in X_{π_{1}(X_{i},x)} concerns only finite subsets of π_{1}(X, x). Hence f(x) is determined by f_{i}(x). If x = x^{∗}, c_{x} is ⋓(c_{x_{i}} : i ∈ I) under the identification of π_{1}(X, x^{∗}) and × ×^{σ}(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I). Then, f is an homeomorphism by Theorem 5.4.

In the case of π_{1}(⋁_{i∈I}(X_{i}, x_{i}), x^{∗}), the difference lies only on x^{∗} and c_{x^{∗}} is ⋓^{F}(c_{x_{i}} : i ∈ I) under the identification of π_{1}(X, x^{∗}) and ∗(π_{1}(X_{i}, x_{i}), c_{x_{i}} : i ∈ I). Then, f is an homeomorphism by Theorem 5.5. □

Corollary 6.7. Let X be a space in the class T . Then X is homeomorphic to X_{π_{1}(X)}.

Corollary 6.8. Let X be a space in the class T . Then, a point in X_{π_{1}(X)} is an algebraic cut point if and only if it is a cut point.

Proof. Byn Theorem 6.6, a cut point is an algebraic cut point. An algebraic cut point is a cut point by Theorem 5.3. □

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School of Science and Engineering, Waseda University, Tokyo 169-8555, JAPAN

  E-mail address: eda@logic.info.waseda.ac.jp