The word problem for some uncountable groups given by countable words

TL;DR

This paper explores the fundamental and homology groups of Griffiths' space and the Hawaiian Earring using countable reduced tame words, revealing uncountably many elements and embedding rational groups.

Contribution

It introduces a novel method using word transformations to analyze uncountable groups and demonstrates the presence of complex algebraic structures within these topological groups.

Findings

Uncountably many elements in the homology groups represented by infinite concatenations of commutators.

The homology groups contain a direct sum of 2^{}} copies of .

The fundamental group of Griffiths' space contains .

Abstract

We investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application we give a short proof that these homology groups contain the direct sum of 2^{\aleph_0} copies of \mathbb{Q}. Finally, we show that the fundamental group of Griffith's space contains \mathbb{Q}.