Right-angled Coxeter groups with Menger curve boundary

TL;DR

This paper identifies conditions under which hyperbolic right-angled Coxeter groups have boundaries homeomorphic to the Menger curve, linking group properties to topological boundary structures.

Contribution

It provides a sufficient condition for the boundary of such groups to be the Menger curve, applicable to various triangulations of surfaces and higher-dimensional disks.

Findings

Boundary homeomorphic to Menger curve under specified conditions

Applicable to many surface triangulations with boundary

Extends to triangulations of disks in higher dimensions

Abstract

We find a sufficient condition for a nerve of a hyperbolic right-angled Coxeter group, under which the boundary of the group is homeomorphic to the Menger curve. We show that this condition is satisfied by many triangulations of surfaces with boundary and other 2-complexes, as well as by some triangulations of disks $D^n$ for arbitrary $n\geq3$.