Hyperbolic buildings, affine buildings and automatic groups

Abstract

We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely generated group acting co-compactly on such a building is hyperbolic, hence automatic. We turn our attention to affine buildings and consider a group $\Gamma$ which acts simply transitively and in a ``type-rotating'' way on the vertices of a locally finite thick building of type $\tilde A_n$. We show that $\Gamma$ is biautomatic, using a presentation of $\Gamma$ and unique normal form for each element of $\Gamma$, as described in ``Groups acting simply transitively on the vertices of a building of type $\tilde A_n$'' by D.I. Cartwright, to appear, Proceedings of the 1993 Como conference ``Groups of Lie type and their geometries''.