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

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

2010-12-19

arxiv.org/abs/1012.4208v1

Mathematics

Group Theory

PDF-LaTeX, 6 pages with 1 figure

Scientific paper

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.

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