Carl Droms

Graph Groups, Coherence, and Three-Manifolds

1. INTRoOU~T~~N We study groups given by presentations each of whose defining relations is of the form ql= yx for some generators .Y and y. To such a presentation we associate a graph X whose vertices are the generators, two vertices .Y and 1’ being adjacent in X if and only if xy = yx is a defining relation. Given a graph X, we denote by GX the group defined by the presentation associated to X in this way. We call GX a graph group. These groups have been studied by Kim and Roush [S], and by Dicks [3]. In this paper we prove the following:

Subgroups of graph groups

Given a graph X, we define a group GX as follows: GX is generated by the vertices of X, with a defining relation xy = JT for each pair -Y, 2’ of ver- tices joined by an edge of X. A group is called a graph group if it is isomorphic to GX for some graph X. We will not distinguish by notation in what follows between a vertex of X and the corresponding element of GX. We remark that the “extreme” cases of graphs, namely the complete and the completely disconnected graphs, correspond, respectively, to free abelian and free groups. It is well known that any subgroup of a group of either of these types is again of the same type, and it is thus natural, to ask to what extent and in what form this is true of the graph groups that lie “in-between” these extreme cases. In this article we will prove the following:

Infinite-ended groups with planar Cayley graphs

Abstract We find necessary and sufficient conditions for a finitely generated group with more than one end to have a planar Cayley graph.

A complex for right-angled Coxeter groups

We associate to each right-angled Coxeter group a 2-dimensional complex. Using this complex, we show that if the presentation graph of the group is planar, then the group has a subgroup of finite index which is a 3-manifold group (that is, the group is virtually a 3-manifold group). We also give an example of a right-angled Coxeter group which is not virtually a 3-manifold group.

Groups assembled from free and direct products

Abstract Let A be the collection of groups which can be assembled from infinite cyclic groups using the binary operations free and direct product. These groups can be described in several ways by graphs. The group ( Z ∗ Z )×( Z ∗ Z ) has been shown by [1] to have a rich subgroup structure. In this article we examine subgroups of A -groups.

Tree Groups and the 4 String Pure Braid Group

Abstract Given a graph Г, undirected, with no loops or multiple edges, we define the graph group on Г, FГ, as the group generated by the vertices of Г, with one relation xy = xy for each pair x and y of adjacent vertices of Г. In this paper we will show that the unpermuted braid group on four strings is an HNN-extension of the graph group Fs, where S = The form of the extension will resolve a conjecture of Tits for the 4-string braid group. We will conclude, by analyzing the subgroup structure of graph groups in the case of trees, that for any tree T on a countable vertex set, Ft is a subgroup of the 4-string braid group. We will also show that this uncountable collection of subgroups of the 4-string braid group is linear, that is, each subgroup embeds in GL(3, R ), as well as embedding in Aut(F), where F is the free group of rank 2.

The Cayley graphs of Coxeter and Artin groups

We obtain a complete classification of the Coxeter and Artin groups whose Cayley graphs with respect to the standard presentations are planar. We also classify those whose Cayley graphs have planar embeddings in which the vertices have no accumulation points

Cylindric embeddings of Cayley graphs

Abstract An embedding of an infinite Cayley graph in the two–sphere has either one, two, or an infinite number of essential accumulation points of vertices. We obtain a list of group presentations which includes every group possessing a Cayley graph that can be embedded in the two–sphere with two essential accumulation points of vertices.