Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

RAAGedy right-angled Coxeter groups II: in the quasiisometry class of the tree RAAGs

Christopher H. Cashen

TL;DR

This work classifies 2-dimensional right-angled Coxeter groups that are quasiisometric to tree right-angled Artin groups, proving that such RACGs always contain a visible finite-index RAAG subgroup when the quasiisometry to a tree RAAG holds. The authors develop a JSJ-decomposition framework for RAAGs and RACGs, describe the graph-of-cylinders structure, and show that cylinders are virtually $\\mathbb{F}\\times\\mathbb{Z}$ while rigid vertices are virtually $\\mathbb{Z}^2$, with no hanging vertices. A key tool is the Dani–Levcovitz finite-index construction (FIDL--$\\Lambda$), which produces a commuting graph $\\Delta$ whose index-$4$ subgroup $A_\\Delta$ yields a visible tree RAAG inside $W_\\Gamma$. They also clarify how pattern data (plane patterns) interact with quasiisometries, and discuss when planarity is not essential. The results illuminate a rigidity phenomenon: quasiisometry to tree RAAGs enforces a strong, verifiable substructure (a visible index-four RAAG) inside the RACG, advancing understanding of how JSJ-type data controls quasiisometry in RAAG/RACG settings.

Abstract

We classify two-dimensional right-angled Coxeter groups that are quasiisometric to a right-angled Artin group defined by a tree, and show that when this is true the right-angled Coxeter group actually contains a visible finite index right-angled Artin subgroup.

RAAGedy right-angled Coxeter groups II: in the quasiisometry class of the tree RAAGs

TL;DR

This work classifies 2-dimensional right-angled Coxeter groups that are quasiisometric to tree right-angled Artin groups, proving that such RACGs always contain a visible finite-index RAAG subgroup when the quasiisometry to a tree RAAG holds. The authors develop a JSJ-decomposition framework for RAAGs and RACGs, describe the graph-of-cylinders structure, and show that cylinders are virtually while rigid vertices are virtually , with no hanging vertices. A key tool is the Dani–Levcovitz finite-index construction (FIDL--), which produces a commuting graph whose index- subgroup yields a visible tree RAAG inside . They also clarify how pattern data (plane patterns) interact with quasiisometries, and discuss when planarity is not essential. The results illuminate a rigidity phenomenon: quasiisometry to tree RAAGs enforces a strong, verifiable substructure (a visible index-four RAAG) inside the RACG, advancing understanding of how JSJ-type data controls quasiisometry in RAAG/RACG settings.

Abstract

We classify two-dimensional right-angled Coxeter groups that are quasiisometric to a right-angled Artin group defined by a tree, and show that when this is true the right-angled Coxeter group actually contains a visible finite index right-angled Artin subgroup.

Paper Structure

This paper contains 9 sections, 3 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology
  4. Visible RAAG subgroups
  5. CFS graphs
  6. JSJ decompositions
  7. Peripheral patterns
  8. RACGs quasiisometric to tree RAAGs
  9. Further remarks

Key Result

Theorem 2.1

Let $\Gamma$ be an incomplete, triangle-free graph without separating cliques. Suppose $\Lambda$ is a disjoint union of trees $\Lambda_0$ and $\Lambda_1$ in $\Gamma^c$ such that $\Lambda$ spans $\Gamma$ and for each $i$, no two vertices of $\Lambda_i$ are adjacent in $\Gamma$. Let $\Delta$ be the co

Figures (4)

  • Figure 1: A nonplanar, triangle-free graph $\Gamma$ such that $W_\Gamma$ is in the quasiisometry class of the tree RAAGs.
  • Figure 2: Graph $\Gamma$ and graph of cylinders of $W_\Gamma$ in \ref{['ex:nonbasicjsj']}.
  • Figure 3: Two graphs as in \ref{['claim:suspendedsuspended']} with their diagonal graphs containing the geodesic $\{p_0,q_0\}, \{p_1,q_1\},\dots,\{p_n,q_n\}$.
  • Figure 4: Graph and graph of cylinders for \ref{['ex:nonplanar_raag_tree_redux']}.

Theorems & Definitions (15)

  • Theorem 2.1: DanLev24
  • Definition 2.1: cf.CasEdl
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • proof
  • Example 2.1
  • Theorem 3.1
  • proof
  • Claim 3.0.1
  • ...and 5 more