[Paper Review] On the coarse geometry of certain 2-dimensional right-angled Coxeter groups

Hoang Thanh Nguyen, Hung Cong Tran|arXiv (Cornell University)|Dec 4, 2017

Geometric and Algebraic Topology24 references3 citations

TL;DR

This paper classifies the coarse geometry of 2-dimensional right-angled Coxeter groups associated with triangle-free, planar, $Γ$-CFS graphs. It shows that such groups are either virtually Seifert or graph manifold groups (when $Γ$ is $Γ$-CFS), or hyperbolic relative to $Γ$-CFS subgroups (otherwise), leading to a complete quasi-isometry classification and divergence types of linear, quadratic, or exponential.

ABSTRACT

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also investigate strongly quasiconvex subgroups, Morse boundary of right-angled Coxeter groups $G_\Gamma$, and their connection to right-angled Artin groups when $\Gamma$ are $\mathcal{CFS}$.

Motivation & Objective

• To classify the quasi-isometry types of right-angled Coxeter groups $G_\Gamma$ for specific planar graphs $\Gamma$.
• To determine the geometric structure of $G_\Gamma$ when $\Gamma$ is $\mathcal{CFS}$, showing it is virtually a Seifert or graph manifold group.
• To analyze the divergence behavior of $G_\Gamma$ and classify it as linear, quadratic, or exponential based on the $\mathcal{CFS}$ property of $\Gamma$.
• To investigate the relationship between strongly quasiconvex subgroups and the Morse boundary of $G_\Gamma$, particularly in relation to right-angled Artin groups when $\Gamma$ is $\mathcal{CFS}$.

Proposed method

• Use of the $\mathcal{CFS}$ (coarse Fuchsian surface) condition on planar, triangle-free graphs $\Gamma$ to classify the geometric type of $G_\Gamma$.
• Application of relative hyperbolicity theory to show that $G_\Gamma$ is hyperbolic relative to $\mathcal{CFS}$ subgroups when $\Gamma$ is not $\mathcal{CFS}$.
• Employment of quasi-isometry invariants to classify $G_\Gamma$ into distinct geometric classes.
• Use of the Morse boundary and strongly quasiconvex subgroups to analyze the large-scale geometry of $G_\Gamma$.
• Comparison with right-angled Artin groups via structural similarities in the $\mathcal{CFS}$ case.
• Topological and combinatorial analysis of planar graphs with no separating vertices or edges to ensure structural rigidity.

Experimental results

Research questions

• RQ1 When is the right-angled Coxeter group $G_\Gamma$ virtually a Seifert manifold group or a graph manifold group, given that $\Gamma$ is a triangle-free, planar, $\mathcal{CFS}$ graph with no separating vertices or edges?
• RQ2 What is the divergence behavior of $G_\Gamma$ when $\Gamma$ is not $\mathcal{CFS}$, and how does it relate to relative hyperbolicity?
• RQ3 How do strongly quasiconvex subgroups of $G_\Gamma$ relate to the Morse boundary in the $\mathcal{CFS}$ case?
• RQ4 What structural parallels exist between $G_\Gamma$ and right-angled Artin groups when $\Gamma$ is $\mathcal{CFS}$?
• RQ5 What is the complete quasi-isometry classification of $G_\Gamma$ for such $\Gamma$?

Key findings

• When $\Gamma$ is $\mathcal{CFS}$, the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or a graph manifold group.
• For non-$\mathcal{CFS}$ $\Gamma$, $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups.
• The divergence of $G_\Gamma$ is linear, quadratic, or exponential, depending on the $\mathcal{CFS}$ status of $\Gamma$.
• A complete quasi-isometry classification is established for all such $G_\Gamma$ groups.
• Strongly quasiconvex subgroups of $G_\Gamma$ are shown to have well-behaved properties in relation to the Morse boundary.
• The Morse boundary and geometric structure of $G_\Gamma$ exhibit structural similarities to those of right-angled Artin groups when $\Gamma$ is $\mathcal{CFS}$.