Quasi-Isometries for certain Right-Angled Coxeter Groups

Abstract

We construct the JSJ tree of cylinders $T_c$ for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas given for hyperbolic RACGs. Additionally, we prove that $T_c$ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided $K_4$. By use of the structure invariant of $T_c$ introduced by Cashen and Martin, we obtain a quasi-isometry-invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry-invariant in case the JSJ decomposition of the RACG does not have any rigid vertices.

Figures (17)

• Figure 2.1.1: The orange vertices form a cut pair and a cut triple respectively.
• Figure 2.3.3: $\Lambda_c$ is the JSJ graph of cylinders of the RACG $W_\Gamma$ obtained by Theorem \ref{['ToChypRACG']}.
• Figure 3.1.1: The special subgroup generated by the orange cut pair $\{v,x\}$ has three purple common adjacent vertices $\{w,m,y\}$. Thus its commensurator is generated by $\{v,w,m,y,x\}$.
• Figure 3.1.2: The common adjacent vertices $\{c_1, \dots, c_k\}$ of $a$ and $b$ are only adjacent to both $a$ and $b$.
• Figure 3.3.3: $\Lambda_c$ is the JSJ graph of cylinders of the RACG $W_\Gamma$.

Theorems & Definitions (100)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7