A new upper bound for the Asymptotic Dimension of RACGs

TL;DR

This paper establishes a new upper bound for the asymptotic dimension of Right-Angled Coxeter groups based on the clique-connected dimension of their defining graphs, linking geometric properties to graph invariants.

Contribution

The paper introduces a novel upper bound for the asymptotic dimension of RACGs using clique-connected dimension and characterizes when these groups are virtually free.

Findings

Asymptotic dimension of RACGs is bounded above by clique-connected dimension of the defining graph.

RACGs are virtually free if and only if the clique-connected dimension of the graph is 1.

Provides a new geometric-combinatorial link between group properties and graph invariants.

Abstract

Let $W_\Gamma$ be the Right-Angled Coxeter group with defining graph $\Gamma$. We show that the asymptotic dimension of $W_\Gamma$ is smaller than or equal to $dim_{CC}(\Gamma)$, the clique-connected dimension of the graph. As a corollary we show that $W_\Gamma$ is virtually free if and only if $dim_{CC}(\Gamma)=1$.