2-dimensional Coxeter groups are biautomatic

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 ∈ 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 $W=\widetilde {A}_3$ .

DETERMINISTIC AND NON-DETERMINISTIC ASYNCHRONOUS AUTOMATIC STRUCTURES

Following the definition of asynchronous automatic structures in [3], we define non-deterministic asynchronous automatic structures and characterize these in terms of the asynchronous fellow traveller property. We show that any group with a non-deterministic asynchronous automatic structure has an asynchronous automatic structure. Non-deterministic asynchronous automatic structures are a labor saving method of showing that a group has an asynchronous automatic structure. They also allow one to define an equivalence relation on the class of non-deterministic asynchronous automatic structures which descends to the subclasses of deterministic asynchronous automatic structures and synchronous automatic structures.