Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces
The paper is a continuation of Eda's work on algebraic properties of the fundamental groups of wild spaces as published in a number of preceding papers. Such groups are a challenging object to study. Even the fact that the fundamental group of the Hawaiian earring is not free requires a surprisingly long argument. In his previous work Eda has studied quasi-atomic groups and proved that many interesting fundamental groups (including the fundamental group of the Hawaiian earring) have this property. A group \(G\) is quasi-atomic if for every homomorphism \(h: G \to A*B\) the image \(h(G)\) is contained in the free product of a finitely generated subgroup and one of the two factor spaces. The current paper introduces a somehow flexible notion of the atomic property. The author proves that free \(\sigma\)-products of groups have a version of the atomic property, i.e., every homomorphic image of a free \(\sigma\)-product of groups in a free product \(*_i A_i\) is ``essentially'' contained in a conjugate subgroup to some \(A_i\). The free \(\sigma\)-product of groups is a certain group encoding an infinite multiplication and which contains the free product. The fundamental group of the Hawaiian earring is a free \(\sigma\)-product of a countable collection of integer groups. Furthermore, the author proves that the fundamental groups of certain wild spaces have another version of the atomic property: if \(X\) is a sufficiently wild Peano continuum then for every injective homomorphism \(h: \pi_1(X)\to *_j H_j\) the image \(h(\pi_1(X))\) is contained in a conjugate subgroup to some \(H_j\).
scientific article; zbMATH DE number 5950719
scientific article