Word calculus in the fundamental group of the Menger curve
Volume 235 / 2016
Fundamenta Mathematicae 235 (2016), 199-226
MSC: Primary 20F65; Secondary 20E08, 55Q52, 57M05, 55Q07.
DOI: 10.4064/fm918-6-2016
Published online: 9 September 2016
Abstract
The fundamental group of the Menger universal curve is uncountable and not free, although all of its finitely generated subgroups are free. It contains an isomorphic copy of the fundamental group of every one-dimensional separable metric space and an isomorphic copy of the fundamental group of every planar Peano continuum. We give an explicit and systematic combinatorial description of the fundamental group of the Menger universal curve and its generalized Cayley graph in terms of word sequences. The word calculus, which requires only two letters and their inverses, is based on Pasynkov’s partial topological product representation and can be expressed in terms of a variation on the classical puzzle known as the Towers of Hanoi.
