English

Nonhyperbolic Coxeter groups with Menger curve boundary (with erratum)

Group Theory 2020-05-18 v5 Geometric Topology

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)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 nn vertices for n5n\geq 5. The construction depends on a slight extension of Sierpi\'nski's theorem on embedding 11--dimensional planar compacta into the Sierpi\'nski carpet. See the appendix for a brief erratum.

Keywords

Cite

@article{arxiv.1812.04649,
  title  = {Nonhyperbolic Coxeter groups with Menger curve boundary (with erratum)},
  author = {Matthew Haulmark and G. Christopher Hruska and Bakul Sathaye},
  journal= {arXiv preprint arXiv:1812.04649},
  year   = {2020}
}

Comments

15 pages, 2 figures. This updated version contains a brief erratum correcting an error in the previous versions

R2 v1 2026-06-23T06:39:29.677Z