Word calculus in the fundamental group of the Menger curve

Hanspeter Fischer ¹   ; Andreas Zastrow ²

¹ Department of Mathematical Sciences Ball State University Muncie, IN 47306, U.S.A.
² Institute of Mathematics Faculty of Mathematics, Physics and Informatics University of Gdańsk 80-308 Gdańsk, Poland

Fundamenta Mathematicae, Tome 235 (2016) no. 3, pp. 199-226

Résumé

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.

Détail

DOI : 10.4064/fm918-6-2016

Keywords: fundamental group menger universal curve uncountable although its finitely generated subgroups contains isomorphic copy fundamental group every one dimensional separable metric space isomorphic copy fundamental group every planar peano continuum explicit systematic combinatorial description fundamental group menger universal curve its generalized cayley graph terms word sequences word calculus which requires only letters their inverses based pasynkov partial topological product representation expressed terms variation classical puzzle known towers hanoi