2-dimensional Coxeter groups are biautomatic. Issue 2 (April 2022)

• Record Type: Journal Article
• Authors: Munro, Zachary; Osajda, Damian; Przytycki, Piotr
• Is Part Of: Proceedings. Volume 152:Issue 2(2022)
• Page Start: 382
• Page End: 401
• Publication Date: 2022-04
• Subjects: Coxeter group -- building -- biautomatic; 20F55; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2021.11
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : 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 $W=\widetilde {A}_3$ .