Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups

TL;DR

This paper explores the relationship between certain subgraphs and right-angled Artin subgroups within two-dimensional right-angled Coxeter groups, providing an algorithm to identify these subgroups based on graph properties.

Contribution

It establishes a connection between satellite-dismantlability of graphs and the existence of specific subgroups, and introduces an algorithm to find or determine the absence of such subgroups.

Findings

Existence of subgroups linked to satellite-dismantlability.

Algorithm for constructing or deciding the non-existence of subgroups.

Characterization of subgroups in two-dimensional cases.

Abstract

There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (\Gamma) and a subgraph (\Lambda) of its complement, checking some conditions, and, if satisfied, producing a graph (\Delta) such that the right-angled Artin group with presentation graph (\Delta) is a finite index subgroup of the right-angled Coxeter group with presentation graph (\Gamma). They do not tell us how to find (\Lambda), given (\Gamma). We show, in the 2--dimensional case, that the existence of such a (\Lambda) is connected to the graph property of satellite-dismantlabilty of (\Gamma), and we use this to give an algorithm for producing a suitable (\Lambda) or deciding that one does not exist.