ATOMIC PROPERTY OF THE FUNDAMENTAL GROUPS OF THE HAWAIIAN EARRING AND WILD

LOCALLY PATH-CONNECTED SPACES

KATSUYA EDA

Abstract. We strengthen previous results on the fundamental groups of the Hawaiian earring and the wild Peano continua. Let X be a path-connected, locally path-connected, first countable space which is not locally semi-simply connected at any point. If the fundamental group π_{1}(X) is a subgroup of a free product ∗_{j∈J}H_{j}, it is contained in a conjugate subgroup to some H_{j}.

1. Introduction

Until recently the Hawaiian earring had been only a typical example of a non-locally simply connected space [12], [17, p. 59], but now the fundamental group of the Hawaiian earring has called attentions of several authors [2, 8, 9, 18]. We call the fundamental group of the Hawaiian earring as the Hawaiian earring group for short, following [2]. Particularly the Hawaiian earring group played a central role in [8], where it is shown that homomorphic images of the Hawaiian earring group in the fundamental group of a one-dimensional metric space determine a point. Using key lemmas in [8], the authur has proved that the fundamental groups of the one-dimensional Peano continua determine their homotopy types [11]. There we see the reflection of the following result about quasi-atomness.

We call a group G quasi-atomic, if for each homomorphism h from G to the free product A ∗ B of arbitrary groups A and B there exists a finitely generated subgroup A′ of A or B′ of B such that Im(h) is contained in A′ ∗ B or A ∗ B′.

By definition, finitely generated groups and abelian groups are quasiatomic, but free products of infinitely generated groups are not quasiatomic. Every homomorphic image of a quasi-atomic group is also quasi-atomic.

1991 Mathematics Subject Classification. Primary 55Q20, Secondary 55Q70, 57M05, 57M07, 20F34.

Key words and phrases. wild space, fundamental group, Hawaiian earring.

Theorem 1.1. [3, Theorem 4.1] (See also [4].)

Let G_{i} be a finitely generated group for each i ∈ I. Then × ×_{i∈I}^{σ}G_{i} is quasi-atomic. Consequently the Hawaiian earring group is quasiatomic.

A topological counter part to the above is given in the following theorem, where A and B denote arbitrary groups.

Theorem 1.2. [10, Theorem 1.5] Let X be a Peano continuum which is not semi-locally simply connected at any point. For every injective homomorphism h : π_{1}(X, x_{0}) → A ∗ B, there exists a finitely generated subgroup A_{0} of A such that Im(h) ≤ A_{0} ∗ B or there exists a finitely generated subgroup B_{0} of B such that Im(h) ≤ A ∗ B_{0}.

In the present paper we strengthen the above results as follows:

Theorem 1.3. Let G_{i} (i ∈ I) and H_{j} (j ∈ J) be groups and h : × ×_{i∈I}^{σ}G_{i} → ∗_{j∈J}H_{j} be a homomorphism to the free product of groups H_{j} s. Then there exists a finite subset F of I and j ∈ J such that h(× ×_{i∈I\F}^{σ} G_{i}) is contained in a subgroup which is conjugate to H_{j}.

Theorem 1.4. Let X be a path-connected, locally path-connected, first countable space which is not semi-locally simply connected at any point and h : π_{1}(X, x_{0}) → ∗_{j∈J}H_{j} be an injective homomorphism. Then the image of h is contained in a conjugate subgroup to some H_{j}.

Corollary 1.5. Let X be a Peano continuum which is not semi-locally simply connected at any point and h : π_{1}(X, x_{0}) → ∗_{j∈J}H_{j} be an injective homomorphism. Then the image of h is contained in a conjugate subgroup to some H_{j}.

Since Theorem 1.4 does not presume compactness, the result can be applied to asymptotic cones of certain kind, since asymptotic cones of word-metric spaces are path-connected, geodesic metric spaces (see Addendum of [1, p.544]). “ The atomic property” in the title refers to this type of general property. It may not be adequate to single out a specific property for the name “atomic property”, since many similar, but slightly distinct variants could be observed in various situations. Theorems 1.1 and 1.2 play important role in [8], [11] and [3], where the information of a wild point of a space is recovered from the fundamental group by these properties. We sum up these properties as atomic property.

A modified form of Theorem 3.1, will be applied in our forth coming paper [5]. There we construct a space from a manifold by attaching copies of the Hawaiian earring and then recover the manifold from the fundamental group of the constructed space according to the construction of spaces from groups in [3].

2. Word theoretic arguments

Since our argument requires results of [7] in detail, we review and reprove some results of [7] and [4] for the reader’s convenience.

Let G_{i} (i ∈ I) be groups such that G_{i} ∩ G_{j} = {e} for distinct i and j. A letter is a non-identity element of ⋃_{i∈I} G_{i}. Two letters are of the same kind, if they belong to the same G_{i}. A word W is a function W : \overline{W} → ⋃_{i∈I} G_{i} such that \overline{W} is a linearly ordered set, W (α) is a letter for each α ∈ \overline{W} and \{α ∈ \overline{W} : W (α) ∈ G_{i}\} is finite for each i ∈ I. A word W is called a σ-word, if \overline{W} is countable. We denote the set of all words by W(G_{i} : i ∈ I) and the set of all σ-words by W^{σ}(G_{i} : i ∈ I).

For a word W ∈ W(G_{i} : i ∈ I) let W^{−} be the word defined by the following: For α ∈ \overline{W} there associates a formal symbol α^{−} with the order such that α^{−} ≤ β^{−} if and only if β ≤ α. Let \overline{W^{−}} = \{α^{−} : α ∈ \overline{W} \} and define W^{−}(α^{−}) = W (α)^{−1} for each α ∈ \overline{W}.

Two words U and V are the same word, denoted by U ≡ V, if there exists an order preserving bijection φ : \overline{U} → \overline{V} such that U(α) = V (φ(α)) for all α ∈ \overline{U}. The notation U = V means that the words U and V present the same element in the inverse limit \underleftarrow{lim}(∗_{i∈F} G_{i}, p_{F F′} : F ⊆ F′ ⋐ I). Here F′ ⋐ I means that F′ is a finite subset of I and p_{F F′} : ∗_{i∈F′}G_{i} → ∗_{i∈F} G_{i} is the projection. The free σ-product × ×_{i∈I}^{σ}G_{i} is the subgroup of \underleftarrow{lim}(∗_{i∈F} G_{i}, p_{F F′} : F ⊆ F′ ⋐ I) consisting of all elements which are presented by σ-words.

The notion of reduced words is defined in [2, 7] so that it coincides with the standard notion for words of finite length [14, 16]. In particular every word has the unique reduced word. A word W is quasi-reduced if the reduced word of W is obtained by multiplying contiguous elements which are the same kind so that the multiplication does not cause cancellation. A word W is cyclically reduced, if W is empty, a single letter, or W W is reduced.

For a word W and a non-negative integer n, we define W^{0} to be an empty word, W^{n+1} ≡ W^{n}W and W^{−(n+1)} ≡ W^{−n}W^{−}.

For a word W and a letter g, let a_{W} (g) be the cardinality of the set \{α ∈ \overline{W} : W (α) = g\}, that is, the number of appearances of g in W.

For a ∈ ∗_{j∈J}H_{j}, l(a) denotes the length of the reduced word for a. Under this convention l(W) is the length of the reduced word of W, but not the length of W itself.

Let W and X be words and g be a letter. The head of W is the appearance of g in gX, if W ≡ gX. Similarly the tail of W is the appearance of g in Xg, if W ≡ Xg. We say that an appearance of a letter g is stable in XgY, if the reduced word of XgY is of the form X′gY′ where X′ and Y′ are the reduced words of X and Y respectively. We simply say that the head of W is stable when it is stable in W, and similarly for the tail. (In the present paper the notions “heads” and “tails” are considered only for non-empty words of finite length.)

The following lemma are for infinitary words, but will be applied for words of finite length.

Lemma 2.1. [7, Corollary 1.6] Let U be a non-empty reduced word such that U ≡ U^{−}. Then there exist a letter u and a word W such that u^{2} = e and U ≡ W^{−}uW.

Lemma 2.2. Let U be a non-empty word such that UU ≡ XU^{−}Y for some words X and Y. Then there exist U_{0} and U_{1} such that U ≡ U_{0}U_{1}, U_{0} ≡ U_{0}^{−} and U_{1} ≡ U_{1}^{−}.

Proof. We have U_{0} and U_{1} such that U_{1}U_{0} ≡ U^{−}, U ≡ XU_{1} ≡ U_{0}Y by assumption. Then U ≡ U_{0}^{−}U_{1}^{−}, which implies U_{0} ≡ U_{0}^{−} and U_{1} ≡ U_{1}^{−}. □

Lemma 2.3. [7, Lemma 2.2] Let H_{j}(j ∈ J) be groups and U and X be reduced words. If the head g or the tail g^{−1} in XUX^{−} is not stable, then l(XUX^{−}) ≤ l(U) + 1.

Proof. It suffices to deal with the case that the tail g^{−1} is not stable. Let V be the reduced word of XU. Then, l(V) ≤ l(X)+l(U). Since the tail g^{−1} in V X^{−} is not stable, l(V X^{−}) ≤ l(V) − l(X^{−}) + 1. Therefore, l(XUX^{−}) ≤ l(U) + 1. □

Lemma 2.4. [7, Lemma 2.3] Let H_{j} (j ∈ J) be groups and m, n, k ∈ ℕ such that m + n + 2 ≤ k. Also let y_{i}, z ∈ ∗_{j∈J}H_{j} (1 ≤ i ≤ M) be elements of the free product of H_{j}. If the element u = y_{1}z^{k} ⋯ y_{M}z^{k} satisfies l(u) ≤ m and l(y_{i}) ≤ n for all 1 ≤ i ≤ M, then one of the following holds:

(1) z is a conjugate to an element of some H_{j};

(2) there exist j, j′ ∈ J, i ∈ {2, ⋯, M}, f ∈ H_{j}, g ∈ H_{j′}, x, y ∈ ∗_{j∈J}H_{j} and a non-negative integer r such that f^{2} = g^{2} = e, z = x^{−1}fxy^{−1}gy, and y_{i} = z^{r}x^{−1}fx or y_{i} = y^{−1}gyz^{r}.

Proof. It is easy to see that there exist reduced words U and W such that z = W^{−}UW and the both words UU and W^{−}UW are quasireduced or UU = e. If l(U) ≥ 2, we can take the above U and W so that UU is reduced. If l(U) ≤ 1, the proof is done. Hence, we assume l(U) ≥ 2 and so also assume that UU is reduced. Let Y_{i} be the reduced word for y_{i} for each 1 ≤ i ≤ M. Then,

u = Y_{1}W^{−}U^{k}W Y_{2}W^{−} ⋯ W Y_{M}W^{−}U^{k}W.

If M = 1, then u = y_{1}z^{k} and hence 2k ≤ k l(z) ≤ l(u) + l(y_{1}) ≤ m + n, which is a contradiction. Hence M ≥ 2. Let p be the least number so that 2p ≥ n + 1. (For later use, p is is taken larger than the present necessity.) Since W^{−}U^{k} is reduced and l(Y_{1}) ≤ n, the reduced word of Y_{1}W^{−}U^{k} is of the form Z_{1}U^{p+2} whether the tail of Y_{1}W^{−} are stable or not. Let Z_{i} be the reduced word of W Y_{i}W^{−} for each 2 ≤ i ≤ M. Now we have

u = Z_{1}U^{p+2}Z_{2}U^{k} ⋯ U^{k}Z_{M}U^{k}W.

We are concerned with the reduced word of U^{p+2}Z_{i}U^{p+2}. Suppose that the head and the tail of U^{p+2}Z_{i}U^{p+2} are stable for all i ≥ 2. Since 2p + 4 ≤ n + 6 ≤ 2k, by considering the rightmost U^{k} we conclude k + (k − n − 2) ≤ 2k − p − 2 ≤ l(u), which is a contradiction. Therefore at least one of the head and the tail of U^{p+2}Z_{i}U^{p+2} is not stable. We fix such an i. If the head and the tail of W Y_{i}W^{−} are stable, U^{p+2}Z_{i}U^{p+2} is a reduced word and hence the head or the tail of W Y_{i}W^{−} is not stable. We have l(Z_{i}) = l(W Y_{i}W^{−}) ≤ l(Y_{i}) + 1 ≤ n + 1 by Lemma 2.3.

The reduced word of Z_{i}U^{p+2} is of form Z′_{i}X′_{i}U^{q+2}, where q ≥ 0, X_{i}X′_{i} ≃ U and l(Z′_{i}) ≤ l(Z_{i}). Now the head or the tail of U^{p+2}Z′_{i}X′_{i}U^{q+2} are not stable. (Case 1): The tail of U^{p+2}Z′_{i}X′_{i}U^{q+2} is not stable.

We examine the cancellation that occurs in the rightmost U^{2}. Then there are S_{i}, T_{i} such that S_{i}T_{i} ≡ U and S_{i} ≃ S_{i}^{−} and T_{i} ≡ T_{i}^{−}. Since UU is reduced, neither S_{i} nor T_{i} is empty. By Lemma 2.1, S_{i} = x_{0}^{−1}fx_{0} and T_{i} = y_{0}^{−1}gy_{0} for some f ∈ H_{j} and g ∈ H_{j′} with f^{2} = g^{2} = e and x_{0}, y_{0} ∈ ∗_{j∈J}H_{j}. Let x = W^{−}x_{0}W and y = W^{−}y_{0}W. Then z = x^{−1}fxy^{−1}gy. Moreover W Y_{i}W^{−}U^{p−q} = Z′_{i}X′_{i} = U^{−j}T_{i} for some j. Hence

y_{i} = Y_{i} = W^{−}U^{−j}T_{i}U^{−(p−q)}W = W^{−}(T_{i}S_{i})^{j}T_{i}(T_{i}S_{i})^{p−q}W.

Now the last term is equal to z^{p−q−j−1}x^{−1}fx if p − q > j and y_{i} = y^{−1}gyz^{j−p+q} otherwise, i.e. p − q ≤ j. (Case 2): The head of U^{p+2}Z′_{i}X′_{i}U^{q+2} is not stable.

We observe the cancellations in the left most U^{2}. Since l(Z′_{i}) ≤ 2p ≤ l(U^{p}), we conclude the existence of S_{i}, T_{i} similarly as the above and get the conclusion. □

Let A and B be groups, and let C_{1} and C_{2} be subsets of A∗B defined by: C_{1} = {x^{−1}ux : u ∈ A ∪ B, x ∈ A ∗ B} and C_{2} = {xy : x, y ∈ C_{1}}.

Note C_{2} is closed under conjugacy, that is, u^{−1}xu ∈ C_{2} if and only if x ∈ C_{2}. We consider cyclically reduced words for elements in C_{2}. A word U is cyclically equivalent to a word V, if U is equivalent to X^{−}V X for some X, i.e. U presents an element conjugate to the element presented by V. Now we easily have:

Lemma 2.5. Every word W for an element of C_{2}(⊆ A∗B) is cyclically equivalent to a word which has one of the following forms:

(1) empty;

(2) u_{0} where u_{0} ∈ A ∪ B;

(3) V_{0}^{−}u_{0}V_{0}v_{0} where u_{0}, v_{0} ∈ A ∪ B and V_{0} is a reduced word.

We remark the following: if W ∈ C_{2} \ C_{1} for a cyclically reduced word W, then W is of the form W_{0}^{−}w_{0}W_{0}W_{1}^{−}w_{1}W_{1}.

In the remaining part of this section A and B are groups. We prove lemmas which have word theoretic characters and will be used in the proof of Theorem 3.1.

Lemma 2.6. Suppose that every element of A has order 2. Let a_{1}, a_{2} ∈ A (a_{1} ≠ a_{2}) and b ∈ B be nontrivial elements. Then

u^{−1}a_{1}uv^{−1}bvu^{−1}a_{2}uv^{−1}bvu^{−1}(a_{1}a_{2})uv^{−1}bv

does not belong to C_{2} for any u, v ∈ A ∗ B.

Proof. Let a_{3} = a_{1}a_{2}. Since

vu^{−1}a_{1}uv^{−1}bvu^{−1}a_{2}uv^{−1}bvu^{−1}(a_{1}a_{2})uv^{−1}bvv^{−1}
= w^{−1}a_{1}wbw^{−1}a_{2}wbw^{−1}a_{2}wbw^{−1}a_{3}wb

where w = uv^{−1}, we may assume v = e and moreover that U^{−}a_{1}UbU^{−}a_{2}UbU^{−}(a_{1}a_{2})Ub is cyclically reduced for the reduced word U for u. Let V ≡ U^{−}a_{1}UbU^{−}a_{2}UbU^{−}a_{3}Ub. We remark a_{3} ≠ a_{1} and a_{3} ≠ a_{2}. Then a_{V} (a_{1}), a_{V} (a_{2}) and a_{V} (a_{3}) are odd, because each of a_{1}, a_{2} nd a_{3} appearing in U also appears in U^{−}. Hence we have three distinct letters g for which a_{V} (g) is odd and g^{2} = e

Since V is cyclically reduced, V does not belong to C_{1}. If V ∈ C_{2}, then by the remark preceding Lemma V must be of the form W_{0}^{−}w_{0}W_{0}W_{1}^{−}w_{1}W_{1}. But then we have at most two distinct letters g for which a_{V} (g) is odd and g^{2} = e. Hence we conclude V ∉ C_{2}. □

Lemma 2.7. Let a ∈ A be an element satisfying a^{2} ≠ e and b ∈ B be a non-trivial element. Then u^{−1}auv^{−1}bvu^{−1}auv^{−1}bvu^{−1}auv^{−1}bv does not belong to C_{2} for every u, v ∈ A ∗ B.

Proof. Let U be the reduced word for u. As in the proof of Lemma 2.6, we may assume that v = e and W ≡ U^{−}aUbU^{−}aUbU^{−}aUb is cyclically reduced. Since the length of W is even, W does not belong to C_{1}. We see a_{W} (a) − a_{W} (a^{−1}) = 3, but if W is of the form W_{0}^{−}w_{0}W_{0}W_{1}^{−}w_{1}W_{1}, a_{W} (g) − a_{W} (g^{−1}) ≤ 2 for every letter g. Hence we conclude W ∉ C_{2} again by the remark preceding to Lemma 2.6. □

Lemma 2.8. Let H be a subgroup of A ∗ B containing ⟨W^{−}aW ⟩ ∗ ⟨V ⟩, where a ∈ A∪B, W^{−}aW is a reduced word and V is a cyclically reduced word with l(V) ≥ 2. There exist u ∈ H such that u ∉ C_{2}.

Proof. Since V is cyclically reduced and l(V) ≥ 2, either W^{−}aW V or V W^{−}aW is reduced. Since the argument proceeds symmetrically, we assume that V W^{−}aW is reduced. Choose k so that k · l(V) > l(W^{−}aW). Then the head of V W^{−}aW is stable in V W^{−}aW V^{k+1}. To show it by contradiction, suppose that it is unstable. Considering the cancellation that occurs in the left most V in V W^{−}aW V^{k+1}, we find that the cancellation does not stop by the multiplication in A or B. To show it by contradiction, suppose that the cancellation stops by the multiplication of c and c′. Then, we have V_{0} and V_{1} such that cV_{0}V_{1} ≡ V ≡ V_{0}^{−}c′V_{1}^{−}. It implies c′ = c^{−1}, which contradicts cc′ ≠ e. Hence we have V_{0} and V_{1} such that V_{0}V_{1} ≡ V, V_{0}^{−} ≡ V_{0}, V_{1}^{−} ≡ V_{1} and W^{−}aW ≡ V_{0}^{−}(V^{−})^{l} for some l ≥ 0. This implies (W^{−}aW V^{l+1})^{2} = (V_{0}V)^{2} = (V_{1})^{2} = e by Lemma 2.1, which contradicts that ⟨W^{−}aW ⟩ and ⟨V ⟩ have no relation.

Let W_{0} be the reduced word of V W^{−}aW V^{k+1}. Since the head and tail of V W^{−}aW V^{k+1} are stable, the word V W_{0}V is reduced. The reduction process of V W^{−}aW V^{k+1} continues until when only the multiplication in A or B appears in the resulting word. By looking at the final stage, we have a non-negative integer l, letters u_{0}, u of the same kind and words U_{0}, U_{1}, X such that U_{0}u_{0}U_{1} ≡ V, W_{0} ≡ XuU_{1}V^{l}, and u ≠ u_{0}.

Let U ≡ W_{0}V^{2k+2}W_{0}V^{6k+8}W_{0}V^{14k+18} which is the reduced word of

V W^{−}aW V^{k+1}V^{2k+2}V W^{−}aW V^{k+1}V^{6k+8}V W^{−}aW V^{k+1}V^{14k+18}.

To show U ∉ C_{2} by contradiction, suppose U ∈ C_{2}. Since U is cyclically reduced, U does not belong to C_{1} and hence is of the form X_{0}^{−}x_{0}X_{0}X_{1}^{−}x_{1}X_{1}. Remark the inequality l(W_{0}) < (2k + 2)l(V), then we see that x_{1} is located in V^{14k+18}. (Case 1) The right most W_{0}V is located in X_{1}^{−}x_{1}X_{1}: Then it is located in X_{1}^{−} because l(V^{14k+17}) > l(W_{0}V^{2k+2}W_{0}V^{6k+8}W_{0}V). Since X_{1} is a subword of V^{14k+18}, we have V_{0} and V_{1} such that V_{0}V_{1} ≡ V, V_{0}^{−} ≡ V_{0}, V_{1}^{−} ≡ V_{1} and W_{0}V ≡ XuU_{1}V^{l}V ≡ Y V_{0}V_{1}V_{0}V^{−l}V_{1} for some word Y.

This implies XuU_{1} ≡ Y V_{0}V_{1} ≡ Y V ≡ Y U_{0}u_{0}U_{1}, which contradicts u ≠ u_{0}. (Case 2) The right most W_{0}V is not located in X_{1}^{−}x_{1}X_{1}: In this case we examine the leftmost W_{0}V and the middle W_{0}V. They are located in X_{0}^{−}, because l(V^{6k+7}) > l(W_{0}V^{2k+2}W_{0}V). Since l(V^{2k+2}) > l(W_{0}), we see the length of the word between the leftmost W_{0} and the middle W_{0} is greater than the length of the rightmost W_{0}. Thus for the leftmost or the middle W_{0}V, a similar argument as Case 1 is used to we deduce a contradiction. □

Lemma 2.9. Let H be a subgroup of A ∗ B such that

(1) H contains a non-trivial element which is conjugate to an element of A or B; and

(2) H is not contained in any conjugate subgroup to A nor B; and (3) H is not contained in any subgroup of the form ⟨u_{0}⟩ ∗ ⟨u_{1}⟩ with

u_{0}^{2} = u_{1}^{2} = e. Then, H contains an element u ∉ C_{2}.

Proof. By the Kurośh subgroup theorem [15], H is of the form ∗_{i∈I}u_{i}^{−1}H_{i}u_{i}∗ ∗_{j∈J}v_{j}^{−1}⟨V_{j}⟩v_{j}, where H_{i} s are subgroups of A or B and V_{j} s are cyclically reduced words and l(V_{j}) ≥ 2. Under the condition (1)-(3) H contains

(a) a subgroup u^{−1}⟨a⟩u∗v^{−1}⟨b⟩v for some non-trivial elements a, b ∈ A ∪ B with a^{2} ≠ e; or

(b) a subgroup ⟨w^{−1}aw⟩ ∗ ⟨v^{−1}V v⟩, where a ∈ A ∪ B and V is a

non-empty cyclically reduced word with l(V) ≥ 2. When (a) holds, Lemma 2.7 implies the conclusion. When (b) holds, vHv^{−1} contains a subgroup ⟨(wv^{−1})^{−1}awv^{−1}⟩ ∗ ⟨V ⟩. Let W be the reduced word for wv^{−1}. If W^{−1}aW is a reduced word, we can apply Lemma 2.8 to vHv^{−1}. Otherwise, W ≡ a_{0}W_{0} with a and a_{0} being the same kind. Let a_{1} = a_{0}^{−1}aa_{0}. Then, a_{1}^{2} = e and W_{0}^{−}a_{1}W_{0} is a reduced word. Hence we can apply Lemma 2.8 to vHv^{−1}. In any case we have an element u ∈ vHv^{−1} satisfying u ∉ C_{2}. We have v^{−1}uv ∈ H and v^{−1}uv ∉ C_{2}. □

Lemma 2.10. Let x ∈ (A ∗ B) \ C_{2}. Then x^{m} ∉ C_{2} for every integer m ≥ 4. For given x_{1}, ⋯, x_{n}, there exists a positive integer m ≥ 4 such that x_{i}x^{m} ∉ C_{1} for every 1 ≤ i ≤ n.

Proof. Let V be the reduced word for x. Obviously l(V) ≥ 2. First we assume that V is cyclically reduced and prove the lemma. To show the first proposition by contradiction, suppose that V^{m} ∈ C_{2}. Then we have letters w_{0}, w_{1} and words W_{0}, W_{1} such that V^{m} ≡ W_{0}^{−}w_{0}W_{0}W_{1}^{−}w_{1}W_{1}.

Since l(V) ≤ l(W_{0}) or l(V) ≤ l(W_{1}), V is a subword of W_{0}^{−} or W_{1}. By Lemma 2.2 we have V_{0}, V_{1} such that V_{0}V_{1} ≡ V, V_{0} ≡ V_{0}^{−} and V_{1} ≡ V_{1}^{−} and consequently V ∈ C_{2}, which is a contradiction.

Next we show the second proposition. Let m_{0} be a natural number such that l(x_{i}) < m_{0}l(V) for every 1 ≤ i ≤ n and m = m_{0} + 3. To show x_{i}V^{m} ∉ C_{1} by contradiction, suppose that x_{i}V^{m} ∈ C_{1}. Then the reduced word for x_{i}V^{m} is of the form XV^{k+3} where l(X) < l(V^{k+1}). Examining the leftmost V, by a similar argument as above we conclude V ∈ C_{2}, which is a contradiction.

In a general case we have u such that the reduced word for u^{−1}xu is cyclically reduced. Since C_{2} is closed under conjugacy, we have the first proposition for x from the corresponding statement for u^{−1}xu. By the cyclically reduced case we choose m for u^{−1}x_{i}u (1 ≤ i ≤ n) so that, for every 1 ≤ i ≤ n, u^{−1}x_{i}x^{m}u = u^{−1}x_{i}u(u^{−1}xu)^{m} ∉ C_{1}. For such m, we have x_{i}x^{m} ∉ C_{1}. □

The following lemma seems to be a folklore, but we prove it for completeness.

Lemma 2.11. Let H be a non-trivial subgroup of ⟨a⟩ ∗ ⟨b⟩ where a^{2} = b^{2} = e.

If H is not conjugate to ⟨a⟩ nor ⟨b⟩, then H contains an element of the form (ab)^{k} for an arbitrarily large even k > 0.

Proof. Every non-empty reduced word of even length is of the form (ab)^{k} or (ba)^{k} for some k > 0 and every reduced word of odd length is of the form W^{−}aW or W^{−}bW for some word W. The reduced word of the concatenation of two words of odd length is of even length and (ba)^{k} is the inverse of (ab)^{k}. Hence, if (ab)^{k} does not belong to H for any even k > 0 and H is not trivial, then H is a conjugate to ⟨a⟩ or ⟨b⟩. In case H contains an element (ab)^{k} for some k > 0, it contains an element (ab)^{k} for arbitrary large even k. □

Lemma 2.12. Let u, w_{0} ∈ ∗_{j∈J}H_{j} and e ≠ h ∈ H_{j_{0}}. If u^{−1}w_{0}^{−1}hw_{0}u ∈ w_{0}^{−1}H_{j_{0}}w_{0}, then u ∈ w_{0}^{−1}H_{j_{0}}w_{0}.

Proof. Under the assumption (w_{0}uw_{0}^{−1})^{−1}hw_{0}uw_{0}^{−1} ∈ H_{j_{0}}. Since H_{j_{0}} is a free factor of the free product, w_{0}uw_{0}^{−1} ∈ H_{j_{0}}, that is, u ∈ w_{0}^{−1}H_{j_{0}}w_{0}.

□

3. Proofs of Theorems 1.3 and 1.4

The following theorem strengthens a part of [3, Theorem 4.1] (see also [4]) and is a special case of Theorem 1.3. In what follows, ℤ_{n} denotes a copy of ℤ indexed by n < ω and a generator of ℤ_{n} is denoted by δ_{n}.

Theorem 3.1. Let A, B be arbitrary groups and h : × ×_{n<ω}ℤ_{n} → A ∗ B be a homomorphism. Then there exist m < ω and u ∈ A ∗ B such that h(× ×_{n≥m}ℤ_{n}) ≤ u^{−1}Au or h(× ×_{n≥m}ℤ_{n}) ≤ u^{−1}Bu.

This section is devoted to a proof of this theorem and those of Theorems 1.3 and 1.4. The first lemma is a very special case of the theorem.

Lemma 3.2. Theorem 3.1 holds, if A = B = ℤ/2ℤ.

Proof. Let a ∈ A and b ∈ B be the non-identity elements. To show the conclusion by contradiction we suppose h(× ×_{n≥m}ℤ_{n}) is not a subgroup of any conjugate of A or B for any natural number m. We construct x_{m} ∈ × ×_{n≥m}ℤ_{n} and positive integers k_{m} by induction. The subgroup h(× ×_{n≥0}ℤ_{n}) contains an element of the form h(x_{0}) = (ab)^{k_{0}} with k_{0} > 0 by Lemma 2.11. For m, we choose x_{m} and even k_{m} so that h(x_{m}) = (ab)^{k_{m}} and k_{m} > Σ_{i=0}^{m−1}k_{i}.

The following construction of an element of × ×_{n<ω}ℤ_{n} is a modification of that in the proofs of [7, Theorem 2.1 and etc.], [6, Theorem 1.1], [3, Theorem 4.1] and [10, Theorem 1.5]. A similar construction will appear in the proof of Theorem 3.1. We start from recalling some notions.

Let Seq be the set of all finite sequences of natural numbers and denote the length of s ∈ Seq by lh(s). An element s ∈ Seq of the length n = lh(s) is written as ⟨s_{0}, ⋯, s_{n−1}⟩ where s_{k} ∈ ℕ (0 ≤ k < n). The lexicographical ordering ≺ on Seq is defined as follows: For s, t ∈ Seq, s ≺ t, if s_{i} < t_{i} for the minimal i with s_{i} ≠ t_{i} or t is an extension of s.

Let W_{m} ∈ W(ℤ_{n} : n ≥ m) be the reduced word for x_{m}, i.e. W_{m} = x_{m}. Let

\overline{V} = \{(s, p) : s ∈ Seq, 0 ≤ s_{i} < k_{i} for 0 ≤ i < lh(s), p ∈ \overline{W_{lh(s)}}\}

be endowed with the lexicographical ordering and define a word V ∈ W(ℤ_{n} : n < ω) by V (s, p) = W_{lh(s)}(p). (We remark h(W_{lh(s)}) = h(x_{lh(s)}) = (ab)^{k_{lh(s)}}.) Then V is a word in W(ℤ_{n} : n < ω). Let

\overline{V_{m}} = \overline{V} ∩ \{s : lh(s) ≥ m, s_{i} = 0 for 0 ≤ i < m\}

and V_{m} be the restriction of V to \overline{V_{m}}. We remark V ≡ V_{0} ≡ (W_{0}V_{1})^{k_{0}} and V_{m} ≡ (W_{m}V_{m+1})^{k_{m}} generally.

We consider h(V) ∈ ⟨a⟩∗⟨b⟩ and take m > 0 so that m > l(h(V)). If l(h(W_{m+1}V_{m+2})) is odd, then h(W_{m+1}V_{m+2})^{2} = e and hence h(V_{m+1}) = h((W_{m+1}V_{m+2})^{k_{m+1}}) = e. Therefore h(V) = (ab)^{k}, where

k = Σ_{i=0}^{m}k_{i}Π_{j=0}^{i}k_{j} ≥ k_{m} ≥ m,

which contradicts m > l(h(V)) = 2k.

Next we assume l(h(W_{m+1}V_{m+2})) is even. Then, h(W_{m+1}V_{m+2}) = (ab)^{p} for some p ≥ 0 or (ba)^{p} for some p > 0. Since h(W_{lh(s)}) = (ab)^{lh(s)}, in the former case we deduce a contradiction similarly to the odd case. In the latter case, we have h(V) = (ba)^{k}, where

k = pk_{m+1}Π_{j=0}^{m}k_{j} − Σ_{i=0}^{m}k_{i}Π_{j=0}^{i}k_{j} ≥ (pk_{m+1} − Σ_{i=0}^{m}k_{i})Π_{j=0}^{m}k_{j} > m,

which contradicts m > l(h(V)) = 2k. Now we have shown the lemma. □

Proof of Theorem 3.1. Let h : × ×_{n<ω}ℤ_{n} → A ∗ B be a homomorphism. We consider the subgroups h(× ×_{n≥m}ℤ_{n}) for m < ω and recall the Kurósh subgroup theorem, i.e a subgroup A ∗ B is of the form ∗_{i∈I}u_{i}^{−1}H_{i}u_{i} ∗ ∗_{j∈J}v_{j}^{−1}⟨V_{j}⟩v_{j}, where H_{i} s are subgroups of A or B and V_{j} s are cyclically reduced and l(V_{j}) ≥ 2. We remark that v_{j}^{−1}⟨V_{j}⟩v_{j} s are free subgroups.

If there exists m < ω such that h(× ×_{n≥m}ℤ_{n}) is contained in a free subgroup, we have m_{0} ≥ m such that h(× ×_{n≥m_{0}}ℤ_{n}) is trivial by the Higman theorem [13] (see also [7, Corollary 3.7]) and we are done. So, we may assume that, for each m, h(× ×_{n≥m}ℤ_{n}) has a free factor u^{−1}Hu for some non-trivial subgroup H of A or B. If, further, there exists m_{0} < ω, u_{0}, u_{1} ∈ A ∪ B and w_{0}, w_{1} ∈ A ∗ B such that u_{0}^{2} = u_{1}^{2} = e, h(× ×_{n≥m}ℤ_{n}) ≤ w_{0}^{−1}⟨u_{0}⟩w_{0} ∗ w_{1}^{−1}⟨u_{1}⟩w_{1}, then the conclusion for w_{0}^{−1}⟨u_{0}⟩w_{0} ∗ w_{1}^{−1}⟨u_{1}⟩w_{1} follows from Lemma 3.2 and so does for A ∗ B.

Therefore, in the following argument we assume that h(× ×_{n≥m}ℤ_{n}) has a free factor u^{−1}Hu for some non-trivia subgroup H of A or B and is not a subgroup of w_{0}^{−1}⟨u_{0}⟩w_{0}∗w_{1}^{−1}⟨u_{1}⟩w_{1} for any m < ω, u_{0}, u_{1} ∈ A∪B with u_{0}^{2} = u_{1}^{2} = e and w_{0}, w_{1} ∈ A ∗ B.

As in the proof of Lemma 3.2 we suppose the negation of the conclusion. Then by Lemma 2.9 we have an element x_{m} ∈ × ×_{n≥m}ℤ_{n} such that h(x_{m}) ∉ C_{2}. Then we choose natural numbers k_{m} by induction. Let k_{0} = 1 and k_{m} be a natural number which meets the following requirements:

(1) k_{m} ≥ 4 and max{l(h(x_{i}^{k_{i}} ⋯ x_{m−1}^{k_{m−1}})) : 0 ≤ i ≤ m−1} + m + 2 ≤ k_{m};

(2) h(x_{i}^{k_{i}} ⋯ x_{m−1}^{k_{m−1}})h(x_{m})^{k_{m}} ∉ C_{1} for every 0 ≤ i ≤ m−1.

The existence of k_{m} and also h(x_{m}^{k_{m}}) ∉ C_{2} are assured by Lemma 2.10. Now we modify the proof of Lemma 3.2. Let W_{m} ∈ W(ℤ_{n} : n ≥ m) be a reduced word for x_{m}^{k_{m}}. Let

\overline{V} = \{(s, p) : s ∈ Seq, 0 ≤ s_{i} < k_{i} for 0 ≤ i < lh(s), p ∈ \overline{W_{lh(s)}}\}

be endowed with the lexicographical ordering and define a word V ∈ W(ℤ_{n} : n ≥ m) by V (s, p) = W_{lh(s)}(p). (We remark h(W_{lh(s)}) = h(x_{lh(s)}^{k_{lh(s)}}).) A subword V_{m} of V is defined by a restriction as before. Let

(1) \overline{V_{m}} = \{(s, p) : s ∈ Seq, lh(s) ≥ m, s_{i} = 0 for 0 ≤ i < m,

0 ≤ s_{i} < k_{i} for m ≤ i < lh(s), p ∈ \overline{W_{lh(s)}}\}

and V_{m} is the restriction of V to \overline{V_{m}}. We remark V_{m} = x_{m}^{k_{m}}V_{m+1}^{k_{m+1}}.

Finally choose m such that l(h(V)) ≤ m. We apply Lemma 2.4 to h(V). Here, n = max{l(h(x_{j}^{k_{i}} ⋯ x_{m−1}^{k_{m−1}})) : 0 ≤ j ≤ m−1}, z = V_{m}, k = k_{m}, M = Π_{j=0}^{m−1}k_{j} and each y_{i} is h(x_{j}^{k_{j}} ⋯ x_{m−1}^{k_{m−1}}) for some 0 ≤ j < m. More precisely, y_{1} = h(x_{0}^{k_{0}} ⋯ x_{m−1}^{k_{m−1}}), y_{2} = h(x_{m−1}^{k_{m−1}}), y_{Π_{j=2}^{m−1}k_{j}} = h(x_{1}^{k_{1}} ⋯ x_{m−1}^{k_{m−1}}) and so on.

When (2) of Lemma 2.4 holds, y_{i} = z^{r}x^{−1}fx or y_{i} = x^{−1}gxz^{r} according to the notation there. Then y_{i} is conjugate to g or f, which implies y_{i} ∈ C_{1} for some 2 ≤ i ≤ M. But, according to our construction h(x_{j}^{k_{j}} ⋯ x_{m−1}^{k_{m−1}}) does not belong to C_{1} for any 0 ≤ j < m. Therefore,

(1) of Lemma 2.4 holds, i.e. V_{m} ∈ C_{1}.

We apply the above argument also to m+1, then we have V_{m+1} ∈ C_{1} and consequently h(x_{m}^{k_{m}}) = h(V_{m}V_{m+1}^{−k_{m+1}}) ∈ C_{2}, which contradicts our construction by Lemma 2.10. □

Now we recall

Lemma 3.3. ( Restricted form of [7, Proposition 1.9]) Let g_{n} (n < ω) be elements of × ×_{i∈I}^{σ}G_{i} such that

{n < ω : a letter in G_{i} appears in the reduced word for g_{n}}

is finite for each i ∈ I.

Then, there exists a homomorphism φ : × ×_{n<ω}ℤ_{n} → × ×_{i∈I}^{σ}G_{i} such that φ(δ_{n}) = g_{n} for n < ω.

Lemma 3.4. [7, Theorem 2.1] Let G_{i} (i ∈ I) and H_{j} (j ∈ J) be groups and h : × ×_{i∈I}^{σ}G_{i} → ∗_{j∈J}H_{j} be a homomorphism to the free product of groups H_{j} s. Then there exist finite subsets F of I and E of J such that h(× ×_{i∈I\F}^{σ} G_{i}) is contained in ∗_{j∈E}H_{j}.

Theorem 1.3 strengthens this result.

Proof of Theorem 1.3. The proof is an application of Theorem 3.1. First we show Theorem 1.3 when J = {0, 1}. Let H_{0} = A and H_{1} = B. To show by contradiction as before, suppose that the conclusion does not hold, that is, for any finite subsets F of I h(× ×_{i∈I\F}^{σ} G_{i}) is not contained in any conjugate of A nor B.

We claim that for each finite subset F of I there exists x ∈ × ×_{i∈I\F}^{σ} G_{i} such that h(x) = w^{−1}aw or w^{−1}bw for some a ∈ A, b ∈ B and w ∈ A∗B. This follows by a similar argument to the first half of the proof of Theorem 3.1. That is, by the Kuroś subgroup Theorem h(× ×_{i∈I\F}^{σ} G_{i}) is a free product of a free group and conjugates of subgroups of A or B. If h(× ×_{i∈I\F}^{σ} G_{i}) is a free group, then by [7, Proposition 3.5] we get the conclusion of the theorem, which contradicts our assumption. Hence, we suppose that h(× ×_{i∈I\F}^{σ} G_{i}) contains a conjugate to a nontrivial subgroup of A or B.

Now we construct x_{m} ∈ × ×_{i∈I}^{σ}G_{i},w_{m} ∈ A ∗ B, u_{m} ∈ A ∪ B and a finite subset F_{m} of I by induction so that the following conditions hold:

(1) x_{m} ∈ × ×_{i∈I^{∗}\F_{m}}^{σ}G_{i} where I^{∗} = ⋃_{m<ω} F_{m} and F_{m} ⊆ F_{m+1};

(2) h(x_{m}) = w_{m}^{−1}u_{m}w_{m} with u_{m} ≠ e;

(3) if both of u_{m} and u_{m+1} belong to A or both of u_{m} and u_{m+1} belong to B, then w_{m} ≠ w_{m+1}, in the other words, if w_{m} = w_{m+1}, then u_{m} ∈ A if and only if u_{m+1} ∈ B.

Along the inductive construction of x_{m}, u_{m} and w_{m}, we choose a countable subset K_{m} of I such that x_{m} ∈ × ×_{i∈K_{m}}^{σ}G_{i} and enumerate K_{m} so that {p(m, n) : n < ω} = K_{m}. Then apply the standard book-keeping method to {p(k, n) : n < ω, k ≤ m} to define F_{m+1} ⊆ ⋃_{k≤m} K_{k}, that is, we let F_{m+1} = {p(k, n) : n ≤ m, k ≤ m}. Under our assumption we continue the construction of x_{m}, u_{m}, w_{m}, K_{m} and F_{m} satisfying (1)-(3).

Let δ_{n} be a generator of ℤ_{n}. By Lemma 3.3 we have a homomorphism φ : × ×_{n<ω}ℤ_{n} → × ×_{i∈I}^{σ}G_{i} such that h(δ_{m}) = x_{m} for m < ω. (The above book-keeping method assures that the sequence (x_{m} : m < ω) satisfies the condition of Lemma 3.3.) By Theorem 3.1 we have n_{0} < ω such that h ∘ φ(× ×_{n≥n_{0}}ℤ_{n}) is contained in a conjugate subgroup to A or B, which contradicts the construction of x_{n_{0}} and x_{n_{0}+1} with (3). Now we have shown the case that J = {0, 1}.

For a general case let h : × ×_{i∈I}^{σ}G_{i} → ∗_{j∈J}H_{j} be a homomorphism. By Lemma 3.4 there exist finite subsets F of I and E of J such that h(× ×_{i∈I\F}^{σ} G_{i}) is contained in ∗_{j∈E}H_{j}. Now the restriction of h to × ×_{i∈I\F}^{σ} G_{i} maps into ∗_{j∈E}H_{j}. Since u^{−1}(∗_{j∈E′}H_{j})u = ∗_{j∈E′}⟨u^{−1}H_{j}u⟩ for E′ ⊆ I, by successive use of the case of J = {0, 1} we have the conclusion. □

Next we prove Theorem 1.4. We recall some notions about loops. For a path f : [0, 1] → X, f^{−} denotes the path defined by: f^{−}(s) = f(1−s) for 0 ≤ s ≤ 1. For paths f : [0, 1] → X and g : [0, 1] → X with f(1) = g(0) we denote the concatenation of the paths f and g by fg. The Hawaiian earring is the plane compactum ℍ = {(x, y) : (x + 1)^{2} + y^{2} = 1/n^{2}, 1≤ n < ω} and each simple closed curve of the Hawaiian earring ℍ is parametrized as follows: e_{n}(t) = ((cos 2πt − 1)/n, sin 2πt/n) for 1 ≤ n < ω, 0 ≤ t ≤ 1. (Here, e_{n} refers to the n-th earring, that is the n-th simple closed curve.) Since π_{1}(ℍ, (0, 0)) is isomorphic to × ×_{n<ω}ℤ_{n} [7, Theorem A.1] and e_{n+1} corresponds to δ_{n} under this isomorphism, we identify δ_{n} and the homotopy class of e_{n+1}. For a path p : [0, 1] → X, we denote the base-point-change isomorphism from π_{1}(X, p(0)) to π_{1}(X, p(1)) by φ_{p}.

Lemma 3.5. Let X be a path-connected, locally path-connected, first countable space which is not semi-locally simply connected at any point and h : π_{1}(X, x_{0}) → ∗_{j∈J}H_{j} be an injective homomorphism. For each point x ∈ X and a path p from x to x_{0} there exists a path-connected open neighborhood U of x satisfying: there exist w_{x} ∈ ∗_{j∈J}H_{j} and j(x) ∈ I such that for every loop l in U with base point x, h ∘ φ_{p}([l]) ∈ w_{x}^{−1}H_{j(x)}w_{x}.

Proof. Let {U_{n} : n < ω} be a neighborhood base of x consisting of path-connected open sets such that U_{n+1} ⊆ U_{n}. To show this by contradiction suppose that such a neighborhood does not exist for a point x and a path p. We inductively construct loops l_{n} in U_{n} with base point x as follows.

Let l_{0} be an essential loop in U_{0} with base point x. Suppose that we have constructed a loop l_{n−1} in U_{n−1} with base point x. If h∘φ_{p}([l_{n−1}]) belongs to a conjugate subgroup to some H_{j}, then we have w_{n−1} ∈ ∗_{j∈J}H_{j} such that h ∘ φ_{p}([l_{n−1}]) ∈ w_{n−1}^{−1}H_{j}w_{n−1}. By the assumption we have an essential loop l_{n} in U_{n} with base point x such that h ∘ φ_{p}([l_{n−1}]) ∉ w_{n−1}^{−1}H_{j}w_{n−1}. Otherwise, we choose an arbitrary essential loop l_{n} in U_{n} with base point x.

Since the images l_{n}([0, 1]) converge to x, we can define a continuous map f : ℍ → X so that f((0, 0)) = x and f ∘ e_{n} = l_{n}. By Theorem 1.3 there exists m such that h∘φ_{p} ∘f_{∗}(× ×_{n≥m}ℤ_{n}) is contained in a subgroup conjugate to some H_{j}. Then, h ∘ φ_{p} ∘ f_{∗}(δ_{m}) belongs to the subgroup conjugate to H_{j}, but h ∘ φ_{p} ∘ f_{∗}(δ_{m+1}) does not belong to the subgroup by our construction, which is a contradiction. □

Proof of Theorem 1.4. Let p be an arbitrary path from x to x_{0}. For each s ∈ [0, 1] let p_{s} be the path defined by: p_{s}(t) = p((1 − s)t). Then p_{s} is a path from p(s) to x_{0}. By Lemma 3.5 we have a connected open neighborhood U_{s} of p(s), w_{s} ∈ ∗_{j∈J}H_{j} and j_{s} ∈ J such that, for any loop l in U_{s} with base point p(s), h ∘ φ_{p_{s}}([l]) ∈ w_{s}^{−1}H_{j_{s}}w_{s}. Since p_{1} is a constant path, for any loop l in U_{1} with base point x_{0} = p(1), h([l]) is contained in w_{1}^{−1}H_{j_{1}}w_{1}.

Considering an open interval which contains s and is contained in p^{−1}(U_{s}) for each s, we have 0 = s_{n} ≤ t_{n} ≤ s_{n−1} ⋯ s_{1} ≤ t_{1} ≤ s_{0} = 1 such that p([0, t_{n}]) ⊆ U_{s_{n}} = U_{0}, p([t_{1}, 1]) ⊆ U_{s_{0}} = U_{1} and p([t_{i+1}, t_{i}]) ⊆ U_{s_{i}} for 1 ≤ i ≤ n − 1.

Define paths q_{i} : [0, 1] → X for 1 ≤ i ≤ n and r_{i} : [0, 1] → X for 0 ≤ i ≤ n−1 by: q_{i}(t) = p(s_{i}(1−t)+t_{i}t) and r_{i}(t) = p(t_{i+1}(1−t)+s_{i}t) respectively. Then q_{i} is a path from p(s_{i}) to p(t_{i}) and r_{i} is a path from p(t_{i+1}) to p(s_{i}), which are restrictions of p. We have an essential loop l_{i} in U_{s_{i}} ∩ U_{s_{i−1}} with base point p(t_{i}) for 1 ≤ i ≤ n. Now

φ_{p_{s_{i}}}([q_{i}^{−}l_{i}q_{i}]) = [(q_{i}p_{s_{i}})^{−}l_{i}(q_{i}p_{s_{i}})]
= [p_{t_{i}}^{−}l_{i}p_{t_{i}}]
= φ_{p_{s_{i−1}}}([r_{i−1}^{−}l_{i}r_{i−1}]).

We remark that q_{i}^{−}l_{i}q_{i} is a loop in in U_{s_{i}} with base points p(s_{i}) and r_{i−1}^{−}l_{i}r_{i−1} is a loop in U_{s_{i−1}} with base point p(s_{i−1}).

We start from h([p_{t_{1}}^{−}l_{1}p_{t_{1}}]) ∈ w_{1}^{−1}H_{j_{1})}w_{1}

h([p_{t_{1}}^{−}l_{1}p_{t_{1}}]) = h(φ_{p_{s_{1}}}([q_{1}^{−}l_{1}q_{1}]) ∈ w_{s_{1}}^{−}H_{j_{s_{1}}}w_{s_{1}}.

Since h is injective and l_{1} is essential, h([p_{t_{1}}^{−}l_{1}p_{t_{1}}]) is a nontrivial element and hence w_{s_{1}} = w_{1} and j_{s_{1}} = j_{1}. Generally we have

h(φ_{p_{s_{i−1}}}([r_{i−1}^{−}l_{i}r_{i−1}]) ∈ w_{s_{i−1}}^{−}H_{j_{s_{i−1}}}w_{s_{i}}
= h(φ_{p_{s_{i}}}([q_{i}^{−}l_{i}q_{i}])) ∈ w_{s_{i}}^{−}H_{j_{s_{i}}}w_{s_{i}}

and by the same reasoning as above w_{s_{i}} = w_{s_{i−1}} and j_{s_{i}} = j_{s_{i−1}}. Hence w_{0} = w_{s_{n}} = w_{1} and j_{0} = j_{1}.

We apply this to the case for x = x_{0} and an arbitrary loop p with base point x_{0}. We have an open neighborhood U_{0} of p(0) = x = x_{0}. Choose an essential loop l in U_{0} ∩ U_{1} with base point x_{0}. Then we have

h([p])^{−1}h([l])h([p]) = h([p^{−}lp]) = h(φ_{p}([l])) ∈ w_{1}^{−}H_{j_{1}}w_{1}

and also h([l]) ∈ w_{1}^{−1}H_{j_{1}}w_{1} by the choice of U_{1}, which is an non-trivial element. Now Lemma 2.12 implies h([p]) ∈ w_{1}^{−1}H_{j_{1}}w_{1}. □

Remark 3.6. (1) The injectivity of a homomorphism in Theorem 1.4 is essential, which is shown in [10, Remark 4.6]. On the other hand the injectivity can be weakened as follows. For a non-empty open set U and a path from a point p(0) in U to x_{0}, let H_{U}^{p} be a subgroup of π_{1}(X, x_{0}) consisting all [p^{−}lp] s for loops l in U with base point p(0).

The injectivity can be weakened to: h(H_{U}^{p}) is non-trivial for an arbitrary non-empty open set U and a path from a point p(0) in U to x_{0}.

(2) The free σ-product × ×_{i∈I}^{σ}G_{i} is realized as the fundamental group of the one point union of spaces whose fundamental groups are isomorphic to G_{i} s. More precisely, let X_{i} be a space locally strongly contractible at x_{i} with π_{1}(X_{i}, x_{i}) = G_{i} and identify all x_{i} s as one point x^{∗}. Let ˜⋁_{i∈I}(X_{i}, x_{i}) be the space with this base set. The topology of each X_{i} \ {x_{i}} is the same as in X_{i}. A neighborhood of x^{∗} is of the form ⋃_{i∈I} O_{i} where each O_{i} is a neighborhood of x_{i} and all but finite many O_{i} s are the whole spaces X_{i}. Then π_{1}( ˜⋁_{i∈I}(X_{i}, x_{i})) is isomorphic to × ×_{i∈I}^{σ}G_{i} [7, Theorem A.1].

Acknowledgment. The author thanks the referee for his careful reading and suitable suggestions, which makes the present paper readable.

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[18] A. Zastrow, A construction of infinitely generated group that is not a freeproduct of surface groups and abelian groups, but freely acts on an ℝ-tree, Proc. Roy. Soc. Edinburgh Ser. A 128 (1998), 433–445.

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

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