2-dimensional Coxeter groups are biautomatic

Abstract

Let $W$ be a $2$-dimensional Coxeter group, that is, a one with $\frac{1}{m_{st}}+\frac{1}{m_{sr}}+\frac{1}{m_{tr}}\le 1$ 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 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 $W={\tilde{A}}_3$.

18 pages, 13 figures

References in corpus (1)

• Weak Modularity and ${\tilde{A}}_n$ Buildings

Cited by in corpus (1)

• Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity