Skip to main content
×
×
Home

CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

  • Thierry Stulemeijer (a1)
Abstract

Given a locally finite leafless tree $T$ , various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$ . We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$ . This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$ , that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$ , the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

Copyright
Footnotes
Hide All

Postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn.

Footnotes
References
Hide All
1. Abramenko, P. and Nebe, G., Lattice chain models for affine buildings of classical type, Math. Ann. 322(3) (2002), 537562.
2. Borel, A. and Tits, J., Homomorphismes ‘abstraits’ de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499571 (French).
3. Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 21 (Springer, Berlin, 1990).
4. Bruhat, F. and Tits, J., Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5251 (French).
5. Bruhat, F. and Tits, J., Groupes réductifs sur un corps local. II. Schémas en groupes, in Existence d’une donnée radicielle valuée, Publications Mathématiques. Institut de Hautes Études Scientifiques, Volume 60, pp. 197376. (1984) (French).
6. Bruhat, F. and Tits, J., Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France 112(2) (1984), 259301 (French).
7. Bruhat, F. and Tits, J., Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires, Bull. Soc. Math. France 115(2) (1987), 141195 (French, with English summary).
8. Burger, M. and Mozes, S., CAT(-1)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc. 9(1) (1996), 5793.
9. Caprace, P.-E. and Radu, N., Chabauty limits of simple groups acting on trees, preprint, 2016, arXiv:1608.00461.
10. Caprace, P.-E. and Stulemeijer, T., Totally disconnected locally compact groups with a linear open subgroup, Int. Math. Res. Not. IMRN 24 (2015), 1380013829.
11. Conrad, B., Gabber, O. and Prasad, G., Pseudo-reductive Groups, second edition, New Mathematical Monographs, Volume 26 (Cambridge University Press, Cambridge, 2015).
12. Deligne, P., Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, Representations of reductive groups over a local field, pp. 119157 (Travaux en Cours, Hermann, Paris, 1984) (French).
13. Demazure, M. and Gabriel, P., Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, Avec un appendice Corps de classes local par Michiel Hazewinkel, 1970 (French).
14. Fesenko, I. B. and Vostokov, S. V., Local Fields and Their Extensions, second edition, Translations of Mathematical Monographs, Volume 121 (American Mathematical Society, Providence, RI, 2002). With a foreword by Igor R. Shafarevich.
15. Georges, E., surjective map of rings with same dimension, Mathematics Stack Exchange. URL: http://math.stackexchange.com/q/604091 (version: 2013-12-12).
16. Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 361 (French).
17. Kazhdan, D., Representations of groups over close local fields, J. Anal. Math. 47 (1986), 175179.
18. Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P., The Book of Involutions, American Mathematical Society Colloquium Publications, Volume 44 (American Mathematical Society, Providence, RI, 1998). With a preface in French by J. Tits.
19. Lang, S., Algebraic number Theory, second edition, Graduate Texts in Mathematics, Volume 110 (Springer, New York, 1994).
20. Mazurkiewicz, S. and Sierpiński, W., Contribution à la topologie des ensembles dénombrables, Fund. Math. 1(1) (1920), 1727 (fre).
21. Pierce, R. S., Associative Algebras, Graduate Texts in Mathematics, Volume 88 (Springer, New York, 1982). Studies in the History of Modern Science, 9.
22. Pink, R., Compact subgroups of linear algebraic groups, J. Algebra 206(2) (1998), 438504.
23. Radu, N., A classification theorem for boundary 2-transitive automorphism groups of trees, Invent. Math. 209(1) (2017), 160.
24. De La Salle, M. and Tessera, R., Local-to-global rigidity of Bruhat-Tits buildings, Illinois J. Math. 60(3–4) (2016), 641654.
25. Serre, J.-P., Local Fields, Graduate Texts in Mathematics, Volume 67 (Springer, New York, 1979). Translated from the French by Marvin Jay Greenberg.
26. The Stacks Project Authors, Stacks Project, 2016. URL: http://stacks.math.columbia.edu.
27. Stulemeijer, T., Reference for Hensel’s Lemma in Algebraic Geometry, URL:http://mathoverflow.net/q/234709 (version: 2016-03-28).
28. Tits, J., Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, Boulder, CO, pp. 3362 (American Mathematical Society, Providence, RI, 1966).
29. Tits, J., Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics, Volume 386 (Springer, New York, 1974).
30. Tits, J., Reductive groups over local fields, in Automorphic Forms, Representations and L-functions, Proceedings of Symposia in Pure Mathematics, Oregon State Univ., Corvallis, OR, 1977, Volume XXXIII, pp. 2969 (American Mathematical Society, Providence, RI, 1979).
31. Tits, J., Immeubles de type affine, in Buildings and the Geometry of Diagrams (Como, (1984), Lecture Notes in Mathematics, Volume 1181, pp. 159190 (Springer, Berlin, 1986) (French).
32. Weiss, R. M., The Structure of Affine Buildings, Annals of Mathematics Studies, Volume 168 (Princeton University Press, Princeton, NJ, 2009).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed