Nonhyperbolic Coxeter groups with Menger boundary

  • Matthew Haulmark

    Vanderbilt University, Nashville, USA
  • G. Christopher Hruska

    University of Wisconsin–Milwaukee, USA
  • Bakul Sathaye

    Wake Forest University, Winston-Salem, USA
Nonhyperbolic Coxeter groups with Menger boundary cover

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Abstract

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article we produce the first known examples of non-hyperbolic CAT(0) groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on vertices for . The construction depends on a slight extension of Sierpinski’s theorem on embedding 1-dimensional planar compacta into the Sierpinski carpet. We give a simplified proof of this theorem using the Baire category theorem.

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Cite this article

Matthew Haulmark, G. Christopher Hruska, Bakul Sathaye, Nonhyperbolic Coxeter groups with Menger boundary. Enseign. Math. 65 (2019), no. 1/2, pp. 207–220

DOI 10.4171/LEM/65-1/2-6