#### Document Type

Article

#### Publication Date

7-1-1989

#### Publication Title

Proceedings of the American Mathematical Society

#### Abstract

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.

#### Volume

106

#### Issue

3

#### First Page Number

573

#### Last Page Number

578

#### DOI

10.2307/2047406

#### Publisher Statement

First published in Proceedings of the American Mathematical Society in 106(3), published by the American Mathematical Society.

#### Suggested Citation

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