2-dimensional Coxeter groups are biautomatic

TL;DR

This paper proves that 2-dimensional Coxeter groups are biautomatic by demonstrating the regularity of a natural geodesic language and its fellow traveller property, with implications for groups acting on certain buildings.

Contribution

It establishes biautomaticity for 2-dimensional Coxeter groups using geodesic language regularity and fellow traveller property, extending to groups acting on buildings of this type.

Findings

Natural geodesic language is regular for all 2D Coxeter groups.

The geodesic language satisfies the fellow traveller property.

Fellow traveller property fails for tilde;A_3.

Abstract

Let $W$ be a $2$-dimensional Coxeter group, that is, a one with $\frac{1}{m_{st}}+\frac{1}{m_{sr}}+\frac{1}{m_{tr}}\leq 1$ for all triples of distinct $s,t,r\in S$. 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 \'{S}wi\k{a}tkowski, 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 $W=\widetilde{A}_3$.