groups graphs and surfaces
Graphs, Groups and Surfaces
1
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.
1
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, with S = {s1 , s2 , . . . } a subset of the element set of
G. Define S −1 = {s−1 , s−1 , . . . } A word w in S is a finite product a1 · · · an , where for all
1
2
1 ≤ i ≤ n, ai ∈ S ∪ S −1 . We say S generates G if every element of G is a finite product in
S and each element of S is a generator for G. A relation is an equality between two words in S. A relator is a word that equals the identity element of G.
Definition 2.2. Let S be a set and let FS be the free group on S. Let R be a set of words on S. < S|R > id defined as FS quotiented by the normal closure of R. G has presentation
< S|R > if G is isomorphic to < S|R >.
Definition 2.3. For every group presentation there is an associated Cayley graph Λ(G, S):
• Every vertex is labelled an element of G.
• Each edge is
1
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.
1
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, with S = {s1 , s2 , . . . } a subset of the element set of
G. Define S −1 = {s−1 , s−1 , . . . } A word w in S is a finite product a1 · · · an , where for all
1
2
1 ≤ i ≤ n, ai ∈ S ∪ S −1 . We say S generates G if every element of G is a finite product in
S and each element of S is a generator for G. A relation is an equality between two words in S. A relator is a word that equals the identity element of G.
Definition 2.2. Let S be a set and let FS be the free group on S. Let R be a set of words on S. < S|R > id defined as FS quotiented by the normal closure of R. G has presentation
< S|R > if G is isomorphic to < S|R >.
Definition 2.3. For every group presentation there is an associated Cayley graph Λ(G, S):
• Every vertex is labelled an element of G.
• Each edge is
References: [1] A. White, Graphs, Groups and Surfaces, Elsevier, New York, 1984. [2] L. Beineke and R. Wilson, Topological Group Theory, Selected Topics in Graph Theory 1 (1978), 15 –49. [3] J. Youngs, Minimal Imbeddings and the Genus of a Graph, Indiana Univ. Math. J. 12 (1963), no. 2, 303 –315. 16