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

We give a short proof of the following theorem of Sang-hyun Kim: if $A(\mathrm{\Gamma })$ is a right-angled Artin group with defining graph $\mathrm{\Gamma }$, then $A(\mathrm{\Gamma })$ contains a hyperbolic surface subgroup if $\mathrm{\Gamma }$ 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.