Let W be a 2-dimensional Coxeter group, that is, one with 1/mst + 1/msr + 1/mtr ≤ 1 for all triples of distinct s, t, r ∈ 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 Świątkowski, 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 $W=\widetilde {A}_3$.

中文翻译：

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二维 Coxeter 组是双自动的

令W是一个二维 Coxeter 群，即对于所有不同的 s、t、r ∈ S的三元组，具有 1/ m st + 1/ m sr + 1/ m tr ≤ 1 的群。我们证明W是双自动的。我们通过证明自然测地线语言是规则的（对于任意W）并满足同伴的属性来做到这一点。因此，通过 Jacek Świątkowski 的工作，在W型建筑物上正确和紧凑地行动的团体也是双自动的。我们还表明，自然语言的同路人属性失败了$W=\widetilde {A}_3$。