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

Christopher H. Cashen, Alexandra Edletzberger

TL;DR

This work tackles when a 2-dimensional right-angled Coxeter group $W_\Gamma$ is commensurable with a RAAG by leveraging Dani–Levcovitz conditions encoded in a two-component subgraph $\Lambda$ of the complement $\Gamma^c$. The authors establish that the existence of such a $\Lambda$ is equivalent to a satellite-dismantlability property of $\Gamma$, and they introduce a constructive Global Search Algorithm along with a Coning Algorithm that builds $\Lambda$ or proves nonexistence. Central contributions include a precise convexity framework for $\Lambda$-hulls, a reduction of the problem to bipartite $(r/b)$-colored components, and a pathway from combinatorial graph properties to finite-index RAAG subgroups inside $W_\Gamma$ via a visual RAAG $A_\Delta$. The work provides algorithmic tools for detecting when RACGs admit RAAG subgroups of finite index, with potential implications for understanding JSJ decompositions and quasi-isometry classifications in the RACG/RAAG landscape.

Abstract

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

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

TL;DR

This work tackles when a 2-dimensional right-angled Coxeter group is commensurable with a RAAG by leveraging Dani–Levcovitz conditions encoded in a two-component subgraph of the complement . The authors establish that the existence of such a is equivalent to a satellite-dismantlability property of , and they introduce a constructive Global Search Algorithm along with a Coning Algorithm that builds or proves nonexistence. Central contributions include a precise convexity framework for -hulls, a reduction of the problem to bipartite -colored components, and a pathway from combinatorial graph properties to finite-index RAAG subgroups inside via a visual RAAG . The work provides algorithmic tools for detecting when RACGs admit RAAG subgroups of finite index, with potential implications for understanding JSJ decompositions and quasi-isometry classifications in the RACG/RAAG landscape.

Abstract

There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (Γ) and a subgraph (Λ) of its complement, checking some conditions, and, if satisfied, producing a graph (Δ) such that the right-angled Artin group with presentation graph (Δ) is a finite index subgroup of the right-angled Coxeter group with presentation graph (Γ). They do not tell us how to find (Λ), given (Γ). We show, in the 2--dimensional case, that the existence of such a (Λ) is connected to the graph property of satellite-dismantlabilty of (Γ), and we use this to give an algorithm for producing a suitable (Λ) or deciding that one does not exist.
Paper Structure (10 sections, 16 theorems, 5 equations, 6 figures)

This paper contains 10 sections, 16 theorems, 5 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. RACGs and RAAGs
  5. Dani-Levcovitz conditions
  6. Splittings of RACGs
  7. FIDL--Lambda Convexity
  8. Satellite-dismantlability and the Coning Algorithm
  9. A search algorithm
  10. Performance

Key Result

Theorem 2.4

If $\Gamma$ is an incomplete, triangle--free graph without separating cliques such that $W_\Gamma$ is quasiisometric to a RAAG then $\Gamma$ is $\mathcal{CFS}$.

Figures (6)

  • Figure 2.1: A bicycle wheel
  • Figure 2.2:
  • Figure 4.1: Example of a Coning Sequence
  • Figure 4.2: An example $T$, $\Theta(\Gamma,\Lambda)$, and $\Delta$ (purple) sitting in $\boxslash(\Gamma)$.
  • Figure 6.1: Graph for Example \ref{['ex:ordermatters']}.
  • ...and 1 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Theorem 2.4: DaniThomasDiv
  • Lemma 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • Theorem 2.8: edletzberger2021quasi
  • ...and 31 more