word metric

Once you can walk around a group via its Cayley graph, you can measure distances: how many moves does it take to get from one element to another? The word metric answers exactly this. The distance from the identity to an element g is the fewest generators (and inverses) you must multiply to spell g — its word length. The distance between g and h is the word length of g^(-1)h, the shortest route from g to h.

Fix a group G and a symmetric generating set S. The word length |g|_S is the minimal n such that g = s_1 s_2 ... s_n with each s_i in S, and |1|_S = 0. Then d_S(g, h) = |g^(-1)h|_S defines a genuine metric on G — symmetric, satisfying the triangle inequality, vanishing only on equal elements. It is precisely the path metric of the Cayley graph Cay(G, S) when each edge is assigned length 1, and it is left-invariant: d_S(xg, xh) = d_S(g, h).

Different generating sets give different word metrics, but for a finitely generated group any two are bi-Lipschitz equivalent: there is a constant C with (1/C)d_S ≤ d_T ≤ C d_S. So while exact distances are convention-dependent, the metric is canonical up to quasi-isometry, which is why coarse geometric invariants of (G, d_S) are honest invariants of the group itself.