[Groupes linéaires discrets contenant des groupes arithmétiques]
If $H$ is a simple real algebraic subgroup of real rank at least two in a simple real algebraic group $G$, we prove, in a substantial number of cases, that a Zariski dense discrete subgroup of $G$ containing a lattice in $H$ is a lattice in $G$. For example, we show that any Zariski dense discrete subgroup of $\mathrm{SL}_n(\mathbb{R})$ ($n\ge 4$) which contains $\mathrm{SL}_3(\mathbb{Z})$ (in the top left hand corner) is commensurable with a conjugate of $\mathrm{SL}_n(\mathbb{Z})$. In contrast, when the groups $G$ and $H$ are of real rank one, there are lattices $\Delta $ in a real rank one group $H$ embedded in a larger real rank one group $G$ and which extends to a Zariski dense discrete subgroup $\Gamma $ of $G$ of infinite co-volume.
Si $H$ est un sous-groupe algébrique réel simple, de rang réel au moins $2$, d’un groupe algébrique réel simple $G$, nous démontrons, dans un bon nombre de cas, qu’un sous-groupe discret Zariski dense de $G$ intersectant $H$ en un réseau, est déjà un réseau dans $G$. Par exemple, nous montrons que tout sous-groupe discret Zariski dense de $\mathrm{SL}_n(\mathbb{R})$ ($n \ge 4$) qui contient $\mathrm{SL}_3(\mathbb{Z})$ (dans le coin supérieur gauche) est commensurable à un conjugué de $\mathrm{SL}_n(\mathbb{Z})$. En revanche, lorsque les groupes $G$ et $H$ sont de rang réel $1$, il existe des réseaux $\Delta $ dans un groupe $H$ de rang réel $1$, plongés dans un plus grand groupe $G$ de rang réel $1$, qui s’étendent en un sous-groupe discret Zariski dense $\Gamma $ de $G$ de co-volume infini.
Accepté le :
Publié le :
Keywords: Lie groups, arithmetic groups, lattices in Lie groups, superrigidity
Mots-clés : Groupes de Lie, groupes arithmétiques, réseaux dans les groupes de Lie, superrigidité
Indira Chatterji 1 ; T. N. Venkataramana 2
CC-BY 4.0
@article{JEP_2025__12__1605_0,
author = {Indira Chatterji and T. N. Venkataramana},
title = {Discrete linear groups containing arithmetic~groups},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1605--1632},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.318},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.318/}
}
TY - JOUR AU - Indira Chatterji AU - T. N. Venkataramana TI - Discrete linear groups containing arithmetic groups JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1605 EP - 1632 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.318/ DO - 10.5802/jep.318 LA - en ID - JEP_2025__12__1605_0 ER -
%0 Journal Article %A Indira Chatterji %A T. N. Venkataramana %T Discrete linear groups containing arithmetic groups %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1605-1632 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.318/ %R 10.5802/jep.318 %G en %F JEP_2025__12__1605_0
Indira Chatterji; T. N. Venkataramana. Discrete linear groups containing arithmetic groups. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1605-1632. doi: 10.5802/jep.318
[Ben00] - “Automorphismes des cônes convexes”, Invent. Math. 141 (2000) no. 1, p. 149-193 | DOI | MR | Zbl
[Ben04] - “Convexes divisibles. I”, in Algebraic groups and arithmetic, Tata Institue of Fundamental Research, Mumbai, 2004, p. 339-374 | Zbl
[BHC61] - “Arithmetic subgroups of algebraic groups”, Bull. Amer. Math. Soc. 67 (1961), p. 579-583 | DOI | MR | Zbl
[BL00] - “Nonarithmetic superrigid groups: counterexamples to Platonov’s conjecture”, Ann. of Math. (2) 151 (2000) no. 3, p. 1151-1173 | DOI | MR | Zbl
[DGK24] - “Combination theorems in convex projective geometry”, 2024 | arXiv | Zbl
[FK65] - Vorlesungen über die Theorie der automorphen Funktionen. Band I: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, vol. 3, 4, Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1965 | MR
[Hal74] - Measure theory, Graduate Texts in Math., vol. 18, Springer, Cham, 1974 | MR | Zbl
[JM87] - “Deformation spaces associated to compact hyperbolic manifolds”, in Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progress in Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, p. 48-106 | DOI | Zbl
[Mar91] - Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3), vol. 17, Springer-Verlag, Berlin, 1991 | MR | Zbl
[Oh98] - “Discrete subgroups generated by lattices in opposite horospherical subgroups”, J. Algebra 203 (1998) no. 2, p. 621-676 | DOI | MR | Zbl
[Tit66] - “Classification of algebraic semisimple groups”, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), American Mathematical Society, Providence, RI, 1966, p. 33-62 | DOI | Zbl | MR
[Ven87] - “Zariski dense subgroups of arithmetic groups”, J. Algebra 108 (1987) no. 2, p. 325-339 | DOI | MR | Zbl
[Ven93] - “On the arithmeticity of certain rigid subgroups”, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) no. 4, p. 321-326 | MR | Zbl
[Ven95] - “On some rigid subgroups of semisimple Lie groups”, Israel J. Math. 89 (1995) no. 1-3, p. 227-236 | DOI | MR | Zbl
[Zim84] - Ergodic theory and semisimple groups, Monographs in Math., vol. 81, Birkhäuser Verlag, Basel, 1984 | DOI | MR | Zbl
Cité par Sources :