Combinatorial Methods for Detecting Surface Subgroups in 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 $n\ge 5$, where ${\bar{C}}_n$ denotes the complement graph of an $n$-cycle. Furthermore, we give a new proof of Kim's co-contraction theorem.

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