Cayley Graphs and the Word Metric

Drawing the group

Fix a group G and a generating set S (assume S = S⁻¹ for symmetry). The Cayley graph Cay(G, S) has one vertex for each element g ∈ G, and an edge from g to gs for every s ∈ S. Multiplying on the *right* by a generator is one step. The result is a connected graph — connected precisely because S generates — and it is the single most important picture in the subject. The group acts on its own Cayley graph by left multiplication, and this action is by graph automorphisms.

Three Cayley graphs to hold in your head:

  G = Z,    S = {+1, -1}      Cay = the infinite line  ... -2 -1 0 1 2 ...

  G = Z^2,  S = {+/-e1, +/-e2} Cay = the infinite square grid

  G = Z/6Z, S = {+1, -1}      Cay = a hexagon (6-cycle)

  G = F_2 = <a,b|>            Cay = the infinite 4-valent TREE
                              (no loops at all -- there are no relations,
                               so no closed paths except backtracking)

A relator of length n in the presentation = a closed loop of length n in Cay.
No relators  <=>  no loops  <=>  a tree.

The word metric

Now make it a metric space. The word metric d_S(g, h) is the length of the shortest path from g to h in the Cayley graph — equivalently the smallest number of generators you multiply to get from g to h. The word length |g| = d_S(1, g) measures g from the identity. This is a genuine metric: it satisfies the triangle inequality, and crucially it is left-invariant, d_S(g, h) = d_S(xg, xh) for all x, because left multiplication is a symmetry. The group is now a space where you can talk about balls, geodesics, and distance.

1. Pick generators S with S = S⁻¹; this makes d symmetric.
2. Define |g| = least k with g = s₁⋯s_k, each sᵢ ∈ S; set |1| = 0.
3. Set d_S(g, h) = |g⁻¹h|; check the triangle inequality from |xy| ≤ |x| + |y|.
4. Verify left-invariance: d_S(xg, xh) = |(xg)⁻¹(xh)| = |g⁻¹h| = d_S(g, h).