Proceedings of the American Mathematical Society
Given a graph F, define the group Fr to be that generated by the vertices of F, with a defining relation xy = yx for each pair x, y of adjacent vertices of F. In this article, we examine the groups Fr, where the graph F is an n-gon, (n > 4). We use a covering space argument to prove that in this case, the commutator subgroup F.' 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 F for which Fr is a free group.
Servatius, H. J., Droms, C., & Servatius, B. (1989). Surface Subgroups of Graph Groups. Proceedings of the American Mathematical Society, 106(3), 573-578. http://dx.doi.org/10.2307/2047406
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First published in Proceedings of the American Mathematical Society in 106(3), published by the American Mathematical Society.