Groups graphs and surfaces

Graphs, Groups and Surfaces Introduction In this paper, we will discuss the interactions among graphs, groups and surfaces. For any given graph, we know that there is an automorphism group associated with it. On the other hand, for any group, we could associate with it a graph representation, namely a Cayley graph of presentations of the group. We will first describe such a correspondence. Also, a graph is always embeddable in some surface. So we will then focus on properties of graphs in terms of their relation to surfaces. Thus, by using the Cayley graphs to describe a group, we can talk about the embeddability of a group.In this way, we see that we can talk about the geometries of a group by looking at their Cayley graphs. Another useful geometric tool to analyze groups is the Dehn diagram. Therefore, in the last section, we will give some comments on how graph theory may be helpful to Dehn diagrams of Coxeter groups. 2 Cayley Graph of Group Presentations In this section we will s ee how Cayley graphs correspond to a particular presentation of a group and how the properties of a group are reflected in the Cayley graphs. Definition 2. 1. Let G be a group