Embeddings of locally compact abelian p-groups in Hawaiian groups

We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

On special subgroups of fundamental group

Suppose ? is a nonzero cardinal number, I is an ideal on arc connected Topological space X, and B?I(X) is the subgroup of ?1(X) (the first fundamental group of X) generated by homotopy classes of ?_I loops. The main aim of this text is to study B?I(X)s and compare them. Most interest is in ? ? {?,c} and I ? {Pfin(X), {?}}, where Pfin(X) denotes the collection of all finite subsets of X. We denote B?{?}(X) with B?(X). We prove the following statements: for arc connected topological spaces X and Y if B?(X) is isomorphic to B?(Y) for all infinite cardinal number ?, then ?1(X) is isomorphic to ?1(Y); there are arc connected topological spaces X and Y such that ?1(X) is isomorphic to ?1(Y) but B?(X) is not isomorphic to B?(Y); for arc connected topological space X we have B?(X) ? Bc(X) ? ?1(X); for Hawaiian earring X, the sets B?(X), Bc(X), and ?1(X) are pairwise distinct. So B?(X)s and B?I(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

Torsion-free word hyperbolic groups are noncommutatively slender

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

Inverse limits of finite rank free groups

We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group.