Groups graphs and surfaces

graphical records, Groups and Surfaces door In this paper, we pull up stakes discuss the interactions among graphs, groups and surfaces. For any presumption graph, we know that there is an automorphism group associated with it. On the other(a) hand, for any group, we could associate with it a graph representation, viz. a Cayley graph of presentations of the group. We will first divulge such a correspondence. Also, a graph is ever embeddable in some surface. So we will then focus on properties of graphs in terms of their proportion to surfaces. Thus, by using the Cayley graphs to describe a group, we depose talk about the embeddability of a group.In this way, we watch over that we go off 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 system may be helpful to Dehn diagrams of Coxeter groups. 2 Cayley Graph of Group Presentations In this section we will see 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