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Word calculus in the fundamental group of the Menger curve

Volume 235 / 2016

Hanspeter Fischer, Andreas Zastrow 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.

Authors

  • Hanspeter FischerDepartment of Mathematical Sciences
    Ball State University
    Muncie, IN 47306, U.S.A.
    e-mail
  • Andreas ZastrowInstitute of Mathematics
    Faculty of Mathematics, Physics and Informatics
    University of Gdańsk
    80-308 Gdańsk, Poland
    e-mail

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