Combinatorial Methods for Detecting Surface Subgroups in Right-Angled Artin Groups

TL;DR

This paper presents combinatorial methods to identify surface subgroups within right-angled Artin groups, providing new proofs of existing theorems and expanding understanding of their subgroup structures.

Contribution

It offers a concise proof of Kim's theorem on surface subgroups and introduces a novel proof of the co-contraction theorem for right-angled Artin groups.

Findings

Surface subgroups exist when the defining graph contains certain induced subgraphs.

New combinatorial proofs simplify understanding of subgroup structures.

Enhanced theoretical framework for analyzing right-angled Artin groups.

Abstract

We give a short proof of the following theorem of Sang-hyun Kim: if $A(\Gamma)$ is a right-angled Artin group with defining graph $\Gamma$, then $A(\Gamma)$ contains a hyperbolic surface subgroup if $\Gamma$ contains an induced subgraph $\bar{C}_n$ for some $n \geq 5$, where $\bar{C}_n$ denotes the complement graph of an $n$-cycle. Furthermore, we give a new proof of Kim's co-contraction theorem.