Previous: [[The universal property of the free group]] This is a brief aside to introduce a structure we'll encounter a lot later on, while generators and relations are fresh in our mind. Suppose we have a group $G = \langle g_1, \ldots, g_n | R \rangle$ given by generators and relations. Then the **Cayley graph** of a group is a directed graph where every vertex is an element of the group, and edges are labelled with generators and their inverses. Traversing an edge labelled $g_i$ from a vertex $g$ gets you to the vertex representing $gg_i$. Hence, this is a really compact way of seeing the multiplication in a group, with the caveat that you first have to select a presentation. For example, the dihedral group for squares has a Cayley graph that looks like ![[dihedral-cayley.png]] (image credit: Wikipedia) This corresponds to the standard presentation of $D_8$ via a rotation and a reflection. Note that if you take a *different* generating set for the same group, you could get a different Cayley graph: ![[dihedral-other-cayley.png]] We can describe a Cayley graph using generators, but what about relations? We can recover relations from a Cayley graph by going around cycles. A cycle in a Cayley graph gives us a word in the generators that is the identity, because multiplying by those generators returns us to the same vertex i.e. element in the graph. This gives us an important characterization of Cayley graphs of free groups. Because they have no relations, they can have no cycles. This means that are regular trees, with the degree of each vertex being twice the number of generators. They look like this: ![[Cayley_graph_of_F2.svg.png]] We will study Cayley graphs as topological spaces of their own, not just as abstract graphs. One can also study them as *metric* spaces, and this gives rise to the field called [geometric group theory](https://en.wikipedia.org/wiki/Geometric_group_theory) Next: [[The free product of groups]]