Cayley graph

The Cayley graph of a group \(G\) with respect to a generating set \(S\) is the graph whose vertices are the elements of \(G\) and whose edges connect \(g\) to \(gs\) for each \(g \in G\) and \(s \in S\). It is the canonical geometric representation of a group, transforming algebraic structure into spatial structure. The Cayley graph is not unique — it depends on the choice of generators — but its large-scale geometry (its quasi-isometry class) is an invariant of the group itself.

The Cayley graph is the foundational object of geometric group theory. It allows group-theoretic properties to be studied through metric geometry: the word problem becomes a question about paths in the graph, growth rates become questions about ball volumes, and subgroup structure becomes questions about embedded subspaces. The free group on \(n\) generators has a Cayley graph that is a tree, while the Cayley graph of \(\mathbb{Z}^2\) is the integer lattice.

The Cayley graph is the group forgetting that it is a group. It is the group as a space, and the space reveals what the algebra conceals: which elements are close, which paths are shortest, and which subgroups are geometrically natural. A group is not defined by its multiplication table; it is defined by the shape of its walks.