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Let (W;S) be a Coxeter system such that no two generators inS commute. Assume that the Cayley graph of (W;S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k d = dist(x;y) there are at most two vertices z such that dist(x;z) = k and dist(z;y) = d k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from x and y is at most 3. This means that the group W is hyperbolic in a sense stronger than that of Gromov.

Structure of geodesics in the Cayley graph of infinite Coxeter groups