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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 $W=\widetilde {A}_3$. [ABSTRACT FROM AUTHOR]

This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of n -ary homogeneous monoids (where every side of every relation has fixed length n ). We then introduce a new encoding technique that allows us to extend the result partially to the class of n -ary multihomogenous monoids. [ABSTRACT FROM AUTHOR]

Abstract: A Coxeter system is called two-dimensional if every parabolic subgroup of rank at least 3 is infinite. We apply results of Niblo and Reeves and of B.T. Williams to show that Coxeter groups with two-dimensional systems are biautomatic if the corresponding presentation contains no visible Euclidean “triangle” groups. This provides a partial positive answer to a question posed by Niblo and Reeves in [G. Niblo, L. Reeves, Coxeter groups act on cube complexes, J. Group Theory 6 (3) (2003) 399–413]. [Copyright &y& Elsevier]