Home Isomorphisms of groups of Kac–Moody type over 𝔽2
Article
Licensed
Unlicensed Requires Authentication

Isomorphisms of groups of Kac–Moody type over 𝔽2

  • Sebastian Bischof EMAIL logo
Published/Copyright: October 23, 2025
Journal of Group Theory
From the journal Journal of Group Theory

Abstract

Caprace and Mühlherr (2006) solved the isomorphism problem for Kac–Moody groups of non-spherical type over finite fields of cardinality at least 4. In this paper, we solve the isomorphism problem for groups of Kac–Moody type (e.g. Kac–Moody groups) over F 2 whose type is 2-complete and A ̃ 2 -free.

Award Identifier / Grant number: 57664192

Funding statement: This work was supported by a fellowship of the German Academic Exchange Service (DAAD) via the grant 57664192.

Acknowledgements

I am grateful to Bernhard Mühlherr for drawing my attention to this problem. I thank him and François Thilmany for many helpful discussions on the topic. I also thank Timothée Marquis for valuable remarks on an earlier draft. I thank the referee for helpful comments.

  1. Communicated by: Adrian Ioana

References

[1] P. Abramenko and K. S. Brown, Buildings, Grad. Texts in Math. 248, Springer, New York, 2008. 10.1007/978-0-387-78835-7Search in Google Scholar

[2] S. Bischof, On commutator relations in 2-spherical RGD-systems, Comm. Algebra 50 (2022), no. 2, 751–769. 10.1080/00927872.2021.1967966Search in Google Scholar

[3] S. Bischof, Construction of RGD-systems of type ( 4 , 4 , 4 ) over F 2 , PhD thesis, Justus-Liebig-Universität Giessen, 2023. Search in Google Scholar

[4] S. Bischof, Construction of commutator blueprints, preprint (2024), https://arxiv.org/abs/2407.15506; to appear in J. Aust. Math. Soc. Search in Google Scholar

[5] S. Bischof, On growth functions of Coxeter groups, Proc. Edinb. Math. Soc. (2) 68 (2025), no. 3, 979–993. 10.1017/S0013091525000094Search in Google Scholar

[6] S. Bischof and B. Mühlherr, Isometries of wall-connected twin buildings, Adv. Geom. 23 (2023), no. 3, 371–388. 10.1515/advgeom-2023-0013Search in Google Scholar

[7] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 4–6, Elem. Math. (Berlin), Springer, Berlin, 2002. 10.1007/978-3-540-89394-3Search in Google Scholar

[8] P.-E. Caprace, On 2-spherical Kac–Moody groups and their central extensions, Forum Math. 19 (2007), no. 5, 763–781. 10.1515/FORUM.2007.031Search in Google Scholar

[9] P.-E. Caprace, “Abstract” homomorphisms of split Kac–Moody groups, Mem. Amer. Math. Soc. 198 (2009), no. 924, 1–84. 10.1090/memo/0924Search in Google Scholar

[10] P.-E. Caprace and B. Mühlherr, Isomorphisms of Kac–Moody groups, Invent. Math. 161 (2005), no. 2, 361–388. 10.1007/s00222-004-0430-zSearch in Google Scholar

[11] P.-E. Caprace and B. Mühlherr, Isomorphisms of Kac–Moody groups which preserve bounded subgroups, Adv. Math. 206 (2006), no. 1, 250–278. 10.1016/j.aim.2005.08.008Search in Google Scholar

[12] A. Devillers, B. Mühlherr and H. Van Maldeghem, Codistances of 3-spherical buildings, Math. Ann. 354 (2012), no. 1, 297–329. 10.1007/s00208-011-0733-5Search in Google Scholar

[13] A. A. Felikson, Coxeter decompositions of hyperbolic polygons, European J. Combin. 19 (1998), no. 7, 801–817. 10.1006/eujc.1998.0238Search in Google Scholar

[14] V. G. Kac and D. H. Peterson, On geometric invariant theory for infinite-dimensional groups, Algebraic Groups (Utrecht 1986), Lecture Notes in Math. 1271, Springer, Berlin (1987), 109–142. 10.1007/BFb0079235Search in Google Scholar

[15] B. Mühlherr, H. P. Petersson and R. M. Weiss, Descent in Buildings, Ann. of Math. Stud. 190, Princeton University, Princeton, 2015. 10.1515/9781400874019Search in Google Scholar

[16] R. Steinberg, Lectures on Chevalley Groups, Univ. Lecture Ser. 66, American Mathematical Society, Providence, 2016. 10.1090/ulect/066Search in Google Scholar

[17] J. Tits and R. M. Weiss, Moufang Polygons, Springer Monogr. Math., Springer, Berlin, 2002. 10.1007/978-3-662-04689-0Search in Google Scholar

Received: 2025-04-14
Revised: 2025-08-25
Published Online: 2025-10-23

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 27.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jgth-2025-0061/pdf
Scroll to top button