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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Surface subgroups of graph groups


Authors: Herman Servatius, Carl Droms and Brigitte Servatius
Journal: Proc. Amer. Math. Soc. 106 (1989), 573-578
MSC: Primary 20F32; Secondary 05C25, 20F34, 57M15
DOI: https://doi.org/10.1090/S0002-9939-1989-0952322-9
MathSciNet review: 952322
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0952322-9
Article copyright: © Copyright 1989 American Mathematical Society