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2-dimensional Coxeter groups are biautomatic

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  23 March 2021

Zachary Munro ,
Damian Osajda  and
Piotr Przytycki
Zachary Munro
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada (zachary.munro@mcgill.ca)
Damian Osajda
Affiliation:
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50–384, Wrocław, Poland Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656, Warszawa, Poland (dosaj@math.uni.wroc.pl)
Piotr Przytycki
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada (piotr.przytycki@mcgill.ca)
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Abstract

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 $W=\widetilde {A}_3$.

Keywords

Coxeter groupbuildingbiautomatic

MSC classification

Primary: 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 382 - 401
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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Footnotes

†

Partially supported by NSERC and AMS.

*

Partially supported by (Polish) Narodowe Centrum Nauki, UMO-2018/30/M/ST1/00668.

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