Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces
type: Item
instance of: scholarly article
title: Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces
zbMATH Open document ID: 1229.55008
author: Katsuya Eda
published in: Journal of the Mathematical Society of Japan
publication date: 2011-09-27T00:00:00Z
review text: 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 \(H_j\).
Mathematics Subject Classification ID: 55Q20
Mathematics Subject Classification ID: 55Q70
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 5950719
zbMATH Keywords: wild space
zbMATH Keywords: fundamental group
zbMATH Keywords: Hawaiian earring
zbMATH Keywords: Peano continuum
reviewed by: Ziga Virk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06330769
OpenAlex ID: W2032130104
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DOI: 10.2969/JMSJ/06330769
description: scientific article; zbMATH DE number 5950719