An infinite commutator-product is not automatically trivial

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. Our technique is based on transformations of associated 2-complexes. As a corollary 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 2ℵ0 copies of Q.