Surface Subgroups of Graph Groups

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1989, Vol.106 (3), p.573-578
Main Authors: Servatius, Herman, Droms, Carl, Servatius, Brigitte
Format: Article
Language:English
Subjects:
Algebra
College mathematics
Commutators
Homomorphisms
Mathematical sets
Orientable surfaces
Vertices
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1989-0952322-9