Abstract
We prove that Coxeter groups are biautomatic. From our construction of the biautomatic structure it follows that uniform lattices in isometry groups of buildings are biautomatic.
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02 October 2025
The original online version of this article was revised: the third affiliation of the first author was missing
References
Amrhein, A.: Characterizing biautomatic groups (2021). arXiv:2105.07509
Bahls, P.: Some new biautomatic Coxeter groups. J. Algebra 296(2), 339–347 (2006)
Bowditch, B.H.: Notes on locally spaces. In: Geometric Group Theory (Columbus, OH, 1992), pp. 1–48 (1995)
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
Brink, B., Howlett, R.B.: A finiteness property and an automatic structure for Coxeter groups. Math. Ann. 296(1), 179–190 (1993)
Caprace, P.-E.: Buildings with isolated subspaces and relatively hyperbolic Coxeter groups. Innov. Incid. Geom. 10, 15–31 (2009)
Caprace, P.-E., Mühlherr, B.: Reflection triangles in Coxeter groups and biautomaticity. J. Group Theory 8(4), 467–489 (2005)
Cartwright, D.I., Shapiro, M.: Hyperbolic buildings, affine buildings, and automatic groups. Mich. Math. J. 42(3), 511–523 (1995)
Casselman, W.A.: Machine calculations in Weyl groups. Invent. Math. 116(1–3), 95–108 (1994)
Coxeter, H.S.M.: Discrete groups generated by reflections. Ann. Math. (2) 35(3), 588–621 (1934)
Davis, M.W.: Buildings are . In: Geometry and Cohomology in Group Theory (Durham, 1994), pp. 108–123 (1998)
Davis, M.W.: The Geometry and Topology of Coxeter Groups. London Mathematical Society Monographs Series, vol. 32. Princeton University Press, Princeton (2008)
Davis, M.W., Shapiro, M.D.: Coxeter groups are automatic, 1–16. Ohio State Mathematical Research Institute Preprints, no. 91-15. (1991)
Deodhar, V.V.: A note on subgroups generated by reflections in Coxeter groups. Arch. Math. (Basel) 53(6), 543–546 (1989)
Dos Santos, F.: Garside shadows and biautomatic structures in Coxeter groups (2025). arXiv:2505.21718
Dyer, M.: Reflection subgroups of Coxeter systems. J. Algebra 135(1), 57–73 (1990)
Dyer, M., Hohlweg, C.: Small roots, low elements, and the weak order in Coxeter groups. Adv. Math. 301, 739–784 (2016)
Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett Publishers, Boston (1992)
Gersten, S.M., Short, H.B.: Small cancellation theory and automatic groups. Invent. Math. 102(2), 305–334 (1990)
Gersten, S.M., Short, H.B.: Small cancellation theory and automatic groups. II. Invent. Math. 105(3), 641–662 (1991)
Moussong, G.: Hyperbolic Coxeter Groups. ProQuest LLC, Ann Arbor (1988). Thesis (Ph.D.)–The Ohio State University
Munro, Z., Osajda, D., Przytycki, P.: 2-dimensional Coxeter groups are biautomatic. Proc. R. Soc. Edinb., Sect. A 152(2), 382–401 (2022)
Niblo, G.A., Reeves, L.D.: The geometry of cube complexes and the complexity of their fundamental groups. Topology 37(3), 621–633 (1998)
Niblo, G.A., Reeves, L.D.: Coxeter groups act on cube complexes. J. Group Theory 6(3), 399–413 (2003)
Noskov, G.A.: Combing Euclidean buildings. Geom. Topol. 4, 85–116 (2000)
Parkinson, J., Yau, Y.: Cone types, automata, and regular partitions in Coxeter groups. Adv. Math. 398, Paper No. 108146, 66 (2022)
Ronan, M.: Lectures on Buildings. University of Chicago Press, Chicago (2009). Updated and revised
Świątkowski, J.: Regular path systems and (bi)automatic groups. Geom. Dedic. 118, 23–48 (2006)
Tits, J.: Sur le groupe des automorphismes de certains groupes de Coxeter. J. Algebra 113(2), 346–357 (1988)
Yau, Y.: Automatic structures for Coxeter groups (2021). Thesis (Ph.D.)–University of Sydney
Acknowledgements
We thank Adrien Abgrall, Pierre-Emmanuel Caprace, Chris Hruska, Jingyin Huang, and Zachary Munro for useful discussions. This paper was written during our stay at the Institut Henri Poincaré in Paris, which we thank for the hospitality.
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The original online version of this article was revised: the third affiliation of the first author was missing
Both authors were partially supported by (Polish) Narodowe Centrum Nauki, UMO-2018/30/M/ST1/00668.
Piotr Przytycki was partially supported by NSERC, AMS, and LabEx CARMIN, ANR-10-LABX-59-01.
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Osajda, D., Przytycki, P. Coxeter groups are biautomatic. Invent. math. 242, 627–637 (2025). https://doi.org/10.1007/s00222-025-01365-6
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DOI: https://doi.org/10.1007/s00222-025-01365-6