Surface subgroups of graph groups

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. To each graph F = (V , E), with vertex set V and edge set E, we associate a presentation PIT whose generators are the elements of V, and whose relations are {xy = yxlx , y adjacent vertices of F} . PI' can be regarded as the presentation of a k-algebra kU, of a monoid Mr, or of a group Fr, called a graph group. These objects have been previously studied by various authors [2-8]. Graph groups constitute a subclass of the Artin groups. Recall that an Artin group is defined by a presentation whose relations all take the form xyx... = yxy -, where the two sides have the same length n > 1, and there is at most one such relation for any pair of generators. To each such presentation we can associate a labeled graph F, which has a vertex for each generator, and for each relation xyx... = yxy* , an edge joining x and y and labeled " n ", where n is the length of each side of the relation. Thus, a graph group is an Artin group whose graph has all edges labeled '2'. In the same context, we mention the conjecture of Tits [1], which states that in the Artin group with labeled graph F, the subgroup generated by the squares of the generators is isomorphic to the graph group F 2, where F2 is the subgraph of F consisting of all the vertices, and all edges labeled '2'. This conjecture has been proved by S. Pride [7] in the case that the graph F contains no triangles. For many graphs F it is true that every subgroup of Fr is itself a graph group [4], besides the obvious cases where F is either complete (Fr free Abelian) or Received by the editors May 23, 1988 and, in revised form, July 15, 1988. This paper was presented at the AMS joint meeting in College Park, Maryland, April 1988.. 1980 Mathematics Subject Classification (1985 Revision). Primary 20F05; Secondary 57M05. (D 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page