Given a graph Γ, define the group F_Γ to be that generated by the vertices of Γ, with a defining relation $xy = yx$ for each pair $x,y$ of adjacent vertices of Γ. In this article, we examine the groups F_Γ, where the graph Γ is an n-gon, (n ≥ 4). We use a covering space argument to prove that in this case, the commutator subgroup Fâ_Γ contains the fundamental group of the orientable surface of genus 1 + (n - 4)2n - 3. We then use this result to classify all finite graphs Γ for which Fâ_Γ is a free group.
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Servatius et al. (1989) studied this question.