Surface subgroups of graph groups

Abstract:

Given a graph $\Gamma$, define the group ${F_\Gamma }$ to be that generated by the vertices of $\Gamma$, with a defining relation $xy = yx$ for each pair $x,y$ of adjacent vertices of $\Gamma$. In this article, we examine the groups ${F_\Gamma }$, where the graph $\Gamma$ is an $n$-gon, $(n \geq 4)$. We use a covering space argument to prove that in this case, the commutator subgroup ${F’_\Gamma }$ contains the fundamental group of the orientable surface of genus $1 + (n - 4){2^{n - 3}}$. We then use this result to classify all finite graphs $\Gamma$ for which ${F’_\Gamma }$ is a free group.