Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-20T14:38:50.120Z Has data issue: false hasContentIssue false
  • English
  • Français

CHABAUTY LIMITS OF SIMPLE GROUPS ACTING ON TREES

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Locally compact groups and their algebras

Published online by Cambridge University Press:  06 August 2018

Pierre-Emmanuel Caprace  and
Nicolas Radu
Pierre-Emmanuel Caprace
Affiliation:
UCLouvain, 1348 Louvain-la-Neuve, Belgium ([email protected])
Nicolas Radu
Affiliation:
UCLouvain, 1348 Louvain-la-Neuve, Belgium ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

Keywords

simple grouplocally compact groupChabauty spacegroups acting on trees

MSC classification

Primary: 20E08: Groups acting on trees 20E32: Simple groups 20E42: Groups with a $BN$-pair; buildings 20F65: Geometric group theory 22D05: General properties and structure of locally compact groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1093 - 1120
Check for updates
Copyright
© Cambridge University Press 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

F.R.S.-FNRS Senior Research Associate, supported in part by the ERC (grant no. 278469).

‡

F.R.S.-FNRS Research Fellow.

References

Abramenko, P. and Brown, K. S., Buildings: Theory and Applications, Graduate Texts in Mathematics, vol. 248 (Springer, New York, 2008).Google Scholar
Abramenko, P. and Brown, K. S., Automorphisms of non-spherical buildings have unbounded displacement, Innov. Incidence Geom. 10 (2010), 1–13.Google Scholar
Banks, C. C., Elder, M. and Willis, G. A., Simple groups of automorphisms of trees determined by their actions on finite subtrees, J. Group Theory 18(2) (2014), 235–261.Google Scholar
Bass, H., Covering theory for graphs of groups, J. Pure Appl. Algebra 89(1–2) (1993), 3–47.Google Scholar
Bass, H. and Kulkarni, R., Uniform tree lattices, J. Amer. Math. Soc. 3(4) (1990), 843–902.Google Scholar
Bass, H. and Lubotzky, A., Tree Lattices, Progress in Mathematics, vol. 176 (Birkhäuser Boston, Inc., Boston, MA, 2001). With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits.Google Scholar
Bass, H. and Tits, J., Discreteness criteria for tree automorphism groups, in Tree lattices (ed. Bass, H. and Lubotzky, A.), Progress in Mathematics, Volume 176, pp. 185–212 (Birkhäuser, Boston, 2001). Appendix to the book.Google Scholar
Bourbaki, N., Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles, No. 1306, p. 222 (Hermann, Paris, 1963). (French).Google Scholar
Burger, M. and Mozes, S., Groups acting on trees: from local to global structure, Inst. Hautes Études Sci. Publ. Math. (92) (2000), 113–150.Google Scholar
Caprace, P.-E. and Ciobotaru, C., Gelfand pairs and strong transitivity for Euclidean buildings, Ergodic Theory Dynam. Systems 35(4) (2015), 1056–1078.Google Scholar
Caprace, P.-E. and Monod, N., Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011), 97–128.Google Scholar
Davis, M. W., The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, vol. 32 (Princeton University Press, Princeton, 2008).Google Scholar
De Medts, T., Silva, A. C. and Struyve, Koen, Universal groups for right-angled buildings, Groups Geom. Dyn. 12(1) (2018), 231–287.Google Scholar
Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic Pro-p groups, 2, Cambridge Studies in Advanced Mathematic, Volume 61 (Cambridge University Press, Cambridge, 1999).Google Scholar
Gao, S., Invariant Descriptive Set Theory, Pure and Applied Mathematics (Boca Raton), Volume 293 (CRC Press, Boca Raton, FL, 2009).Google Scholar
Gelanger, T., A lecture on invariant random subgroups, in New Directions in Locally Compact Groups (ed. Caprace, P.-E. and Monod, N.), London Mathematical Society Lecture Note Series, Volume 447, pp. 186–204 (Cambridge University Press, Cambridge, 2018).Google Scholar
Guivarc’h, Y. and Rémy, B., Group-theoretic compactification of Bruhat–Tits buildings, Ann. Sci. Éc. Norm. Supér. 39(6) (2006), 871–920.Google Scholar
Kaniuth, E. and Taylor, K. F., Induced representations of locally compact groups, in Cambridge Tracts in Mathematics, Volume 197 (Cambridge University Press, Cambridge, 2013).Google Scholar
Marquis, T., Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity, preprint, 2015, arXiv:1509.01976.Google Scholar
Mazurkiewicz, S. and Sierpiński, W., Contribution à la topologie des ensembles dénombrables, Fund. Math. 1(1) (1920), 17–27. (French).Google Scholar
Mosher, L., Sageev, M. and Whyte, K., Maximally symmetric trees, Geom. Dedicata 92 (2002), 195–233. Dedicated to John Stallings on the occasion of his 65th birthday.Google Scholar
Radu, N., A classification theorem for boundary 2-transitive automorphism groups of trees, Invent. Math. 209(1) (2017), 1–60.Google Scholar
Serre, J.-P., Arbres, Amalgames SL2, Astérisque, Volume 46 (1977). (French).Google Scholar
Stulemeijer, T., Chabauty limits of algebraic groups acting on trees, preprint, 2016, arXiv:1610.08454.Google Scholar
Tits, J., Sur le groupe des automorphismes d’un arbre, Essays on topology and related topics (Mémoires dédiés à Georges de Rham), pp. 188–211 (Springer, New York, 1970). (French).Google Scholar
Tits, J., Buildings and group amalgamations, in Proceedings of groups—St. Andrews 1985, London Mathematical Society Lecture Note Series, Volume 121, pp. 110–127 (Cambridge University Press, Cambridge, 1986).Google Scholar