a Cayley graph
/ KAY-lee /
How do you draw a picture of a group? The Cayley graph gives a literal one: put a dot for every element of the group, and draw an arrow from each element to the elements you can reach by applying one generator. The result is a network that you can walk around, and the shape of that network is the geometric face of the group. It is the bridge that lets you see an algebraic object as a space.
Concretely, fix a group G and a generating set S. The Cayley graph Cay(G, S) has one vertex for each element g of G, and an edge from g to gs for every generator s in S. Walking along edges from the identity vertex spells out a word in the generators, so paths in the graph are words and the graph encodes the multiplication structure visually. Give every edge length 1 and measure distance by shortest paths: the resulting path metric on vertices is exactly the word metric d_S, so the Cayley graph is the geometric realization of the word metric. The group acts on its own Cayley graph by left multiplication — sending g to ag carries the whole graph to itself isometrically — so the graph looks identical from every vertex, a homogeneous space in the most concrete sense.
Cayley graphs are the standard object of study in geometric group theory: properties of the group become geometric properties of the graph. A group is hyperbolic when its Cayley graph is Gromov-hyperbolic; the growth of the group is the growth of balls in the graph; the number of ends of the group is the number of ends of the graph. The all-important caveat mirrors the word metric: the graph depends on the choice of S, and different generating sets give genuinely different-looking graphs (the integers with one generator give a line, with an extra generator give something denser). What is well-defined independent of S is the quasi-isometry class of the graph, so Cayley graphs are studied up to quasi-isometry, and statements about 'the geometry of the group' always mean coarse, generator-independent features.
The Cayley graph of the free group F_2 = <a, b> on two generators is the infinite 4-valent tree: from every vertex there are four edges (for a, a^{-1}, b, b^{-1}), and because the group is free there are no nontrivial loops, so the graph has no cycles. This tree is 0-hyperbolic, which is the geometric reason F_2 is a hyperbolic group. By contrast the Cayley graph of Z^2 with standard generators is the square grid in the plane, which is flat and not hyperbolic.
F_2's Cayley graph is the 4-valent tree (hyperbolic); Z^2's is the flat square grid.
The Cayley graph depends on the generating set, so its detailed shape is not a group invariant. Only its quasi-isometry class — capturing growth, hyperbolicity, ends — is intrinsic to the group; never read off an exact feature that changes when you change generators.