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$A(\Gamma)$ is a right-angled Artin group with defining graph $\Gamma$, then$A(\Gamma)$ contains a hyperbolic surface subgroup if $\Gamma$ contains aninduced subgraph $\bar{C}_n$ for some $n \geq 5$, where $\bar{C}_n$ denotes thecomplement graph of an $n$-cycle. Furthermore, we give a new proof of Kim'sco-contraction theorem.