Let W be a 2-dimensional Coxeter group, that is, one with 1/m(st) + 1/m(sr) + 1/m(tr) <= 1 for all triples of distinct s, t, r is an element of S. We prove that W is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary W), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Swiatkowski, groups acting properly and cocompactly on buildings of type W are also biautomatic. We also show that the fellow traveller property for the natural language fails for W = (A) over tilde (3).
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