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Right-angled Artin subgroups of right-angled Coxeter and Artin groups

Pallavi Dani, Ivan Levcovitz

Abstract

We determine when certain natural classes of subgroups of right-angled Coxeter groups (RACGs) and right-angled Artin groups (RAAGs) are themselves RAAGs. We characterize finite-index visual RAAG subgroups of 2-dimensional RACGs. As an application, we show that any 2-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it is commensurable to a RAAG. Additionally, we give new examples of RACGs with non-planar defining graphs which are commensurable to RAAGs. Finally, we give a new proof of a result of Dyer: every subgroup generated by conjugates of RAAG generators is itself a RAAG.

Right-angled Artin subgroups of right-angled Coxeter and Artin groups

Abstract

We determine when certain natural classes of subgroups of right-angled Coxeter groups (RACGs) and right-angled Artin groups (RAAGs) are themselves RAAGs. We characterize finite-index visual RAAG subgroups of 2-dimensional RACGs. As an application, we show that any 2-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it is commensurable to a RAAG. Additionally, we give new examples of RACGs with non-planar defining graphs which are commensurable to RAAGs. Finally, we give a new proof of a result of Dyer: every subgroup generated by conjugates of RAAG generators is itself a RAAG.

Paper Structure

This paper contains 12 sections, 37 theorems, 19 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Basic terminology and notation
  4. Right-angled Coxeter and Artin groups
  5. Disk Diagrams
  6. Visual RAAG subgroups of right-angled Coxeter groups
  7. Three or more $\Lambda$-components
  8. Finite index visual RAAGs
  9. Applications
  10. Non-planar RACGs commensurable to RAAGs
  11. $2$-dimensional RACGs with planar defining graph
  12. Generalized reflection subgroups of RAAGs

Key Result

Theorem A

Let $W_\Gamma$ be a $2$-dimensional RACG. Let $\Lambda$ be a subgraph of $\Gamma^c$ with no isolated vertices, and let $G$ be the subgroup generated by $E(\Lambda)$. Then the following are equivalent.

Figures (15)

  • Figure 2.1: A disk diagram in a RACG with boundary the word $s_2 s_3 s_4 s_5 s_3 s_2 s_1 s_6 s_5 s_4 s_6 s_1$ and base vertex $p$. Two hyperplanes are shown in red. As these hyperplanes intersect, it must be that $s_4$ commutes with $s_2$.
  • Figure 3.1:
  • Figure 3.2: In the figure, the colored parts consist of $\Lambda$-edges, and the black parts consist of $\Gamma$-edges. The condition $\mathcal{R}_3$ says that if $\Theta$ contains a black square as shown, then every vertex of $T_c$ is joined by a $\Gamma$-edge to every vertex of $T_d$.
  • Figure 3.3: The graph on the left concerns Example \ref{['ex:R4_necessary']} and the graph of the right concerns Example \ref{['ex:R4_complicated']}.
  • Figure 3.4: This figure illustrates condition $\mathcal{R}_4$. The green subgraph is $T_c$ and the blue subgraph is $T_d$. The condition says that any edge in the 2-component cycle (shown in solid black edges) is part of a square of $\Gamma$ with two vertices in $T_c$ and two in $T_d$. This is illustrated for the edge from $d_3$ to $c_4$. The dotted lines are $\Gamma$-edges which are not necessarily in the 2-component cycle.
  • ...and 10 more figures

Theorems & Definitions (98)

  • Definition 1.1: RAAG system
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D: Dyer
  • Definition 2.1: Deletion Condition
  • Theorem 2.2: basarab
  • proof
  • Definition 2.3: Tits moves
  • Theorem 2.4: Titsbasarab
  • ...and 88 more