Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary

TL;DR

This work addresses the problem of characterizing word hyperbolic Coxeter groups by their Gromov boundary, focusing on when the boundary is the $Sierpiński$ curve or the $Menger$ curve. The authors formulate complete, finite-criteria characterizations in terms of the labelled nerve $L^\bullet$ (and its planarity, separability, and puncture-respecting cohomological dimension $\mathrm{pcd}$), extending Moussong’s hyperbolicity criteria and leveraging Cannon’s conjecture for Coxeter groups. The main results state that $\partial W$ is the $Sierpiński$ curve if and only if $L^\bullet$ is unseparable, planar, and not a $3$-cycle, and that $\partial W$ is the $Menger$ curve if and only if $L^\bullet$ is unseparable, $\mathrm{pcd}(L^\bullet)=1$, and not planar; these conclusions extend to decomposable groups via labelled joins with a simplex. The approach combines topological characterizations of the curves (Whyburn, Anderson), boundary-splitting criteria (Bowditch, Mihalik–Tschantz), and planarity arguments, providing a complete, topology-driven classification for this important family of hyperbolic groups with Coxeter structure.

Abstract

We give complete characterizations (in terms of nerves) of those word hyperbolic Coxeter groups whose Gromov boundary is homeomorphic to the Sierpiński curve and to the Menger curve, respectively. The justification is mostly an appropriate combination of various results from the literature.

Theorems & Definitions (11)

• Theorem 1
• Proposition 1.1: M. Kapovich and B. Kleiner kk
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• Definition 1.5
• Definition 1.6
• Lemma 1.7
• Remark
• Lemma 1.8: J. Świątkowski, Lemma 1.3 in js