Free and non-free subgroups of the fundamental group of the Hawaiian earrings.
type: Item
instance of: scholarly article
title: Free and non-free subgroups of the fundamental group of the Hawaiian earrings.
zbMATH Open document ID: 1064.20031
DOI: 10.2478/BF02475662
author: Andreas Zastrow
published in: Central European Journal of Mathematics
publication date: 2003-05-18T00:00:00Z
review text: The Hawaiian earring \(H=C_1\cup C_2\cup\cdots\) is the compact planar set that is the union of the circles \(C_n\) of radius \(1/n\) and center \((0,-1/n)\). The fundamental group \(G=\pi_1(H,0)\) of \(H\) is an uncountable group that is locally free but not free. Considerable effort has been expended in recent years to show that the group \(G\) nevertheless deserves to be considered free in some generalized sense and to determine which of its many subgroups are free or not free in the classical sense. This paper adds to the pantheon of examples. Most interesting are the following: (1) The first known example of a countable subgroup of \(G\) that is not free. (2) A subgroup that is free, contains the standard ``generators'' (the classes of the simple closed curves in \(H)\), but for which those generators cannot be extended to a free basis. (3) An uncountable free subgroup that differs substantially from all known free subgroups. (4) Examples showing how little one needs to change the description to pass from a free subgroup to a subgroup that is not free.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1862498
zbMATH Keywords: free subgroups
zbMATH Keywords: fundamental groups
zbMATH Keywords: Hawaiian earrings
zbMATH Keywords: word sequences
zbMATH Keywords: almost free subgroups
reviewed by: James W. Cannon
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.2478/bf02475662
OpenAlex ID: W2010763428
description: scientific article; zbMATH DE number 1862498