The noncommutative factor theorem for lattices in product groups
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 513-524.

We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices Γ<G in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras L(Γ)ML(ΓG/P) sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.

Nous montrons un théorème du facteur de Bader-Shalom non commutatif pour les réseaux avec des projections denses dans les groupes produits. Comme application de ce résultat et de nos travaux précédents, nous obtenons un théorème du facteur de Margulis non commutatif pour tous les réseaux irréductibles Γ<G dans les groupes algébriques semi-simples de rang supérieur. En particulier, nous donnons une description complète de toutes les sous-algèbres de von Neumann intermédiaires L(Γ)ML(ΓG/P) situées entre l’algèbre de von Neumann de groupe et l’algèbre de von Neumann d’espace de mesure de groupe associée à l’action sur le bord de Furstenberg-Poisson.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.223
Classification: 22E40, 37A40, 46L10, 46L55
Keywords: Factor theorem, Furstenberg-Poisson boundaries, group measure space von Neumann algebras, higher rank lattices in semisimple algebraic groups
Mot clés : Théorème du facteur, bords de Furstenberg-Poisson, algèbres de von Neumann d’espace de mesure de groupe, groupes algébriques semi-simples de rang supérieur

Rémi Boutonnet 1; Cyril Houdayer 2

1 Institut de Mathématiques de Bordeaux, CNRS, Université Bordeaux I 33405 Talence, France
2 Université Paris-Saclay, Laboratoire de Mathématiques d’Orsay, CNRS, Institut Universitaire de France 91405 Orsay, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Rémi Boutonnet; Cyril Houdayer. The noncommutative factor theorem for lattices in product groups. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 513-524. doi : 10.5802/jep.223. https://jep.centre-mersenne.org/articles/10.5802/jep.223/

[BBH21] U. Bader, R. Boutonnet & C. Houdayer - “Charmenability of higher rank arithmetic groups”, 2021, to appear in Ann. H. Lebesgue | arXiv

[BBHP22] U. Bader, R. Boutonnet, C. Houdayer & J. Peterson - “Charmenability of arithmetic groups of product type”, Invent. Math. 229 (2022) no. 3, p. 929-985 | DOI | MR | Zbl

[BH21] R. Boutonnet & C. Houdayer - “Stationary characters on lattices of semisimple Lie groups”, Publ. Math. Inst. Hautes Études Sci. 133 (2021), p. 1-46 | DOI | MR | Zbl

[BM00a] M. Burger & S. Mozes - “Groups acting on trees: from local to global structure”, Publ. Math. Inst. Hautes Études Sci. 92 (2000), p. 113-150 | DOI | Numdam | Zbl

[BM00b] M. Burger & S. Mozes - “Lattices in product of trees”, Publ. Math. Inst. Hautes Études Sci. (2000) no. 92, p. 151-194 | DOI | Numdam | MR | Zbl

[BS06] U. Bader & Y. Shalom - “Factor and normal subgroup theorems for lattices in products of groups”, Invent. Math. 163 (2006) no. 2, p. 415-454 | DOI | MR | Zbl

[CM11] P.-E. Caprace & N. Monod - “Decomposing locally compact groups into simple pieces”, Math. Proc. Cambridge Philos. Soc. 150 (2011) no. 1, p. 97-128, Corrigendum: Ibid., 164 (2018), no. 2, p. 381–384 | DOI | MR | Zbl

[Fur63a] H. Furstenberg - “Noncommuting random products”, Trans. Amer. Math. Soc. 108 (1963), p. 377-428 | DOI | MR | Zbl

[Fur63b] H. Furstenberg - “A Poisson formula for semi-simple Lie groups”, Ann. of Math. (2) 77 (1963), p. 335-386 | DOI | MR | Zbl

[Fur02] A. Furman - “Random walks on groups and random transformations”, in Handbook of dynamical systems, vol. 1A, North-Holland, Amsterdam, 2002, p. 931-1014 | DOI | Zbl

[Hou21] C. Houdayer - “Noncommutative ergodic theory of higher rank lattices”, 2021 | arXiv

[KP22] M. Kalantar & N. Panagopoulos - “On invariant subalgebras of group C * and von Neumann algebras”, Ergodic Theory Dynam. Systems (2022), First view | DOI

[Mar91] G. A. Margulis - Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3), vol. 17, Springer-Verlag, Berlin, 1991 | DOI

[NZ99] A. Nevo & R. J. Zimmer - “Homogenous projective factors for actions of semi-simple Lie groups”, Invent. Math. 138 (1999) no. 2, p. 229-252 | DOI | MR | Zbl

[Pet14] J. Peterson - “Character rigidity for lattices in higher-rank groups”, 2014, Preprint

[Str74] R. A. Struble - “Metrics in locally compact groups”, Compositio Math. 28 (1974), p. 217-222 | Numdam | MR | Zbl

[Suz20] Y. Suzuki - “Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems”, Comm. Math. Phys. 375 (2020) no. 2, p. 1273-1297 | DOI | MR | Zbl

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