nLab Cayley graph

Idea

The Cayley graph, also called the Cayley quiver, of a group, $G$ with a given set of generators, $X$, encodes how the chosen generating elements operate (by multiplication) on the elements of the group.

Definition

Given a group, $G$, and a set of generators, $X$, which we assume form a subset of $|G|$, we can form a graph with the elements of $G$ as the vertices. The edges of the graph will be directed and will be labelled (some people say ‘coloured’) by the elements of $X$ with an edge joining a vertex, $g$, to the vertex $g x$ labelled by $x$.

This graph is called the Cayley graph of the group, $G$, relative to the set of generators.

Example

Consider one of the standard presentations of $S_3$, $(a,b : a^3, b^2, (a b)^2)$. Write $r = a^3$, $s = b^2$, $t = (ab)^2$.

The Cayley graph is easy to draw. There are two triangles corresponding to $1 \to a \to a^2$ and to its translate by $b$, $b \to a b \to a^2 b$, flipping the orientation of the second, and three 2-cycles, $1\to b\to 1$, $a\to a b\to a$ and $a^2\to a^2b\to a^2$.

To understand the geometric significance of this graph we compare the algebraic information in the presentation with the homotopical information in the graph (considered as a CW-complex).

Looking at the presentation it leads to a free group, $F$, on the generators, $a$ and $b$, so $F$ is free of rank 2, but the normal closure of the relations $N(R)$ is a subgroup of $F$, so it must be free as well, by the Nielsen-Schreier theorem. Its rank will be 7, given by the Schreier index formula.

Looked at geometrically, this will be the fundamental group of the Cayley graph, of the presentation. This group is free on generators corresponding to edges outside a maximal tree, and, of course, there are 7 of these.

References

Last revised on July 15, 2018 at 02:47:28. See the history of this page for a list of all contributions to it.