Volume 6 (2023)
Bader, Uri;  Boutonnet, Rémi;  Houdayer, Cyril
Charmenability of higher rank arithmetic groups
Annales Henri Lebesgue, Volume 6 (2023), pp. 297-330.

Metadata

DOI10.5802/ahl.166
Keywords Arithmetic groups, Characters, Lattices, Poisson boundaries, Simple algebraic groups, von Neumann algebras

Abstract

We complete the study of characters on higher rank semisimple lattices initiated in [BH21, BBHP22], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo–Zimmer structure theorem obtained in [BH21] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis’ factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.


References

[AGV14] Abért, Miklós; Glasner, Yair; Virág, Bálint Kesten’s theorem for invariant random subgroups, Duke Math. J., Volume 163 (2014) no. 3, pp. 465-488 | MR | Zbl

[AM66] Auslander, Louis; Moore, Calvin C. Unitary representations of solvable Lie groups, Memoirs of the American Mathematical Society, 62, American Mathematical Society, 1966 | Zbl

[BBHP22] Bader, Uri; Boutonnet, Rémi; Houdayer, Cyril; Peterson, Jesse Charmenability of arithmetic groups of product type, Invent. Math., Volume 229 (2022) no. 3, pp. 929-985 | DOI | MR | Zbl

[BDL17] Bader, Uri; Duchesne, Bruno; Lécureux, Jean Almost algebraic actions of algebraic groups and applications to algebraic representations, Groups Geom. Dyn., Volume 11 (2017) no. 2, pp. 705-738 | DOI | MR | Zbl

[Bek07] Bekka, Bachir Operator-algebraic superridigity for SL n (ℤ), n≥3, Invent. Math., Volume 169 (2007) no. 2, pp. 401-425 | DOI | MR | Zbl

[BF14] Bader, Uri; Furman, Alex Boundaries, rigidity of representations, and Lyapunov exponents, Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. III: Invited lectures, KM Kyung Moon Sa, Seoul (2014), pp. 71-96 | Zbl

[BF20] Bader, Uri; Furman, Alex Super-rigidity and non-linearity for lattices in products, Compos. Math., Volume 156 (2020) no. 1, pp. 158-178 | DOI | MR | Zbl

[BFGW15] Bader, Uri; Furman, Alex; Gorodnik, Alex; Weiss, Barak Rigidity of group actions on homogeneous spaces. III, Duke Math. J., Volume 164 (2015) no. 1, pp. 115-155 | MR | Zbl

[BH21] Boutonnet, Rémi; Houdayer, Cyril Stationary characters on lattices of semisimple Lie groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 133 (2021), pp. 1-46 | DOI | MR | Zbl

[BH22] Boutonnet, Rémi; Houdayer, Cyril The noncommutative factor theorem for lattices in product groups (2022) (https://arxiv.org/abs/2207.13548v1)

[BN13] Bowen, Lewis; Nevo, Amos Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dyn. Syst., Volume 33 (2013) no. 3, pp. 777-820 | DOI | MR | Zbl

[BO08] Brown, Nathanial P.; Ozawa, Narutaka C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, 2008 | Zbl

[Bor91] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer, 1991 | Zbl

[BS06] Bader, Uri; Shalom, Yehuda Factor and normal subgroup theorems for lattices in products of groups, Invent. Math., Volume 163 (2006) no. 2, pp. 415-454 | DOI | MR | Zbl

[BT71] Borel, Armand; Tits, Jacques Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math., Volume 12 (1971), pp. 95-104 | DOI | Zbl

[CH89] Cowling, Michael; Haagerup, Uffe Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math., Volume 96 (1989) no. 3, pp. 507-549 | DOI | MR | Zbl

[CIOS21] Chifan, Ionut; Ioana, Adrian; Osin, Denis; Sun, Bin Wreath-like product groups and rigidity of their von Neumann algebras (2021) (https://arxiv.org/abs/2111.04708v1)

[Con80] Connes, Alain A factor of type II 1 with countable fundamental group, J. Oper. Theory, Volume 4 (1980), pp. 151-153 | MR | Zbl

[Con94] Connes, Alain Noncommutative geometry, Academic Press Inc., 1994 (Transl. from the French by Sterling Berberian) | Zbl

[CP22] Creutz, Darren; Peterson, Jesse Character rigidity for lattices and commensurators (2022) (To appear in American Journal of Mathematics, https://arxiv.org/abs/1311.4513)

[Fur67] Furstenberg, Hillel Poisson boundaries and envelopes of discrete groups, Bull. Am. Math. Soc., Volume 73 (1967), pp. 350-356 | DOI | MR | Zbl

[GK96] Ge, Liming; Kadison, Richard On tensor products for von Neumann algebras, Invent. Math., Volume 123 (1996) no. 3, pp. 453-466 | MR | Zbl

[GL18] Gelander, Tsachik; Levit, Arie Invariant random subgroups over non-Archimedean local fields, Math. Ann., Volume 372 (2018) no. 3-4, pp. 1503-1544 | DOI | MR | Zbl

[GW15] Glasner, Eli; Weiss, Benjamin Uniformly recurrent subgroups, Recent trends in ergodic theory and dynamical systems. International conference in honor of S. G. Dani’s 65th birthday, Vadodara, India, December 26–29, 2012. Proceedings (Contemporary Mathematics), Volume 631, American Mathematical Society, 2015, pp. 63-75 | MR | Zbl

[Hab78] Haboush, William J. Homogeneous vector bundles and reductive subgroups of reductive algebraic groups, Am. J. Math., Volume 100 (1978), pp. 1123-1137 | DOI | MR | Zbl

[HM79] Howe, Roger E.; Moore, Calvin C. Asymptotic properties of unitary representations, J. Funct. Anal., Volume 32 (1979), pp. 72-96 | DOI | MR | Zbl

[Hou21] Houdayer, Cyril Noncommutative ergodic theory of higher rank lattices (2021) (https://arxiv.org/abs/2110.07708)

[Ioa18] Ioana, Adrian Rigidity for von Neumann algebras, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. III. Invited lectures, World Scientific; Sociedade Brasileira de Matemática (2018), pp. 1639-1672 | Zbl

[IPV13] Ioana, Adrian; Popa, Sorin; Vaes, Stefaan A class of superrigid group von Neumann algebras, Ann. Math., Volume 178 (2013) no. 1, pp. 231-286 | DOI | MR | Zbl

[Izu04] Izumi, Mazaki Non-commutative Poisson boundaries, Discrete geometric analysis. Proceedings of the 1st JAMS symposium, Sendai, Japan, December 12–20, 2002 (Contemporary Mathematics), Volume 347, American Mathematical Society, 2004, pp. 69-81 | MR | Zbl

[LL20] Lavi, Omer; Levit, Arie Characters of the group EL d (R) for a commutative Noetherian ring R (2020) (https://arxiv.org/abs/2007.15547)

[Mar91] Margulis, Grigoriĭ A. Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17, Springer, 1991 | Zbl

[NZ99] Nevo, Amos; Zimmer, Robert J. Homogenous projective factors for actions of semi-simple Lie groups, Invent. Math., Volume 138 (1999) no. 2, pp. 229-252 | DOI | MR | Zbl

[NZ02] Nevo, Amos; Zimmer, Robert J. A structure theorem for actions of semisimple Lie groups, Ann. Math., Volume 156 (2002) no. 2, pp. 565-594 | DOI | MR | Zbl

[Pau02] Paulsen, Vern Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, 78, Cambridge University Press, 2002 | MR | Zbl

[Pet14] Peterson, Jesse Character rigidity for lattices in higher-rank groups (2014) (Preprint, https://math.vanderbilt.edu/peters10/rigidity.pdf)

[Pin98] Pink, Richard Compact subgroups of linear algebraic groups, J. Algebra, Volume 206 (1998) no. 2, pp. 438-504 | DOI | MR | Zbl

[Pop07] Popa, Sorin Deformation and rigidity for group actions and von Neumann algebras, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume I: Plenary lectures and ceremonies, European Mathematical Society, 2007, pp. 445-477 | Zbl

[PT16] Peterson, Jesse; Thom, Andreas Character rigidity for special linear groups, J. Reine Angew. Math., Volume 716 (2016), pp. 207-228 | DOI | MR | Zbl

[Ric77] Richardson, Roger W. Affine coset spaces of reductive algebraic groups, Bull. Lond. Math. Soc., Volume 9 (1977), pp. 38-41 | DOI | MR | Zbl

[Sha99] Shalom, Yehuda Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T), Trans. Am. Math. Soc., Volume 351 (1999) no. 8, pp. 3387-3412 | DOI | MR | Zbl

[Spr98] Springer, Tonny A. Linear algebraic groups, Progress in Mathematics, 9, Birkhäuser, 1998 | DOI | Zbl

[Suz20] Suzuki, Yuhei Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, Commun. Math. Phys., Volume 375 (2020) no. 2, pp. 1273-1297 | DOI | MR | Zbl

[SZ94] Stuck, Garett; Zimmer, Robert J. Stabilizers for ergodic actions of higher rank semisimple groups, Ann. Math., Volume 139 (1994) no. 3, pp. 723-747 | DOI | MR | Zbl

[Tak02] Takesaki, Masimichi Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, 124, Springer, 2002 | Zbl

[Tak03] Takesaki, Masimichi Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, 127, Springer, 2003 | Zbl

[Tit64] Tits, Jacques Algebraic and abstract simple groups, Ann. Math., Volume 80 (1964), pp. 313-329 | DOI | MR | Zbl

[Tit66] Tits, Jacques Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) (Proceedings of Symposia in Pure Mathematics), American Mathematical Society, 1966, pp. 33-62 | Zbl

[Vae10] Vaes, Stefaan Rigidity for von Neumann algebras and their invariants, Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. III: Invited lectures, World Scientific; Hindustan Book Agency, 2010, pp. 1624-1650 | Zbl