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Super-rigidity and non-linearity for lattices in products

Part of: Lie groups Locally compact groups and their algebras

Published online by Cambridge University Press:  26 November 2019

Uri Bader  and
Alex Furman
Uri Bader
Affiliation:
Weizmann Institute, Rehovot, Israel email [email protected]
Alex Furman
Affiliation:
University of Illinois at Chicago, USA email [email protected]
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Abstract

We prove a super-rigidity result for algebraic representations over complete fields of irreducible lattices in products of groups and lattices with dense commensurator groups. We derive criteria for the non-linearity of such groups.

Keywords

super-rigiditylatticesPolish fields

MSC classification

Primary: 22E40: Discrete subgroups of Lie groups
Secondary: 22D40: Ergodic theory on groups
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 1 , January 2020 , pp. 158 - 178
Copyright
© The Authors 2019 

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Footnotes

U. Bader was supported in part by the ISF-Moked grant 2095/15 and the ERC grant 306706. A. Furman was supported in part by the NSF grant DMS 1611765.

References

Bader, U., Caprace, P.-E. and Lécureux, J., On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow , J. Amer. Math. Soc. 32 (2019), 491–562; MR 3904159.Google Scholar
Bader, U., Duchesne, B. and Lécureux, J., Almost algebraic actions of algebraic groups and applications to algebraic representations , Groups Geom. Dyn. 11 (2017), 705–738.Google Scholar
Bader, U. and Furman, A., Algebraic representations of ergodic actions and super-rigidity, Preprint (2014), arXiv:1311.3696.Google Scholar
Bader, U. and Furman, A., Boundaries, rigidity of representations, and Lyapunov exponents , in Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III (Kyung Moon SA, Seoul, 2014), 71–96; MR 3729019.Google Scholar
Bader, U. and Furman, A., An extension of Margulis’ super-rigidity theorem, in Dynamics, geometry and number theory: the impact of Margulis on modern mathematics, Chicago Lectures in Mathematics, (University of Chicago Press), to appear. Preprint (2018), arXiv:1810.01607.Google Scholar
Bader, U., Furman, A. and Sauer, R., Lattice envelopes , Duke Math. J. (2019), to appear. Preprint (2017), arXiv:1711.08410.Google Scholar
Bader, U. and Shalom, Y., Factor and normal subgroup theorems for lattices in products of groups , Invent. Math. 163 (2006), 415–454.Google Scholar
Bernšteĭn, I. N. and Zelevinskiĭ, A. V., Representations of the group GL (n, F), where F is a local non-Archimedean field , Uspekhi Mat. Nauk 31 (1976), 5–70.Google Scholar
Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126, second edition (Springer, New York, 1991).Google Scholar
Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean analysis: a systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261 (Springer, Berlin, 1984); MR 746961.Google Scholar
Breuillard, E. and Gelander, T., A topological Tits alternative , Ann. of Math. (2) 166 (2007), 427–474.Google Scholar
Bruhat, F. and Tits, J., Groupes réductifs sur un corps local , Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251; MR 0327923 (French).Google Scholar
Burger, M. and Monod, N., Continuous bounded cohomology and applications to rigidity theory , Geom. Funct. Anal. 12 (2002), 219–280.Google Scholar
Caprace, P.-E. and Monod, N., Isometry groups of non-positively curved spaces: discrete subgroups , J. Topol. 2 (2009), 701–746.Google Scholar
Effros, E. G., Transformation groups and C ∗ -algebras , Ann. of Math. (2) 81 (1965), 38–55.Google Scholar
Engler, A. J. and Prestel, A., Valued fields, Springer Monographs in Mathematics (Springer, Berlin, 2005).Google Scholar
Gelander, T., Karlsson, A. and Margulis, G. A., Superrigidity, generalized harmonic maps and uniformly convex spaces , Geom. Funct. Anal. 17 (2008), 1524–1550.Google Scholar
Glasner, E. and Weiss, B., Weak mixing properties for non-singular actions , Ergodic Theory Dynam. Systems 36 (2016), 2203–2217.Google Scholar
Kaimanovich, V. A., Double ergodicity of the Poisson boundary and applications to bounded cohomology , Geom. Funct. Anal. 13 (2003), 852–861.Google Scholar
Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156 (Springer, New York, 1995).Google Scholar
Lifschitz, L., Arithmeticity of rank-1 lattices with dense commensurators in positive characteristic , J. Algebra 261 (2003), 44–53.Google Scholar
Margulis, G. A., Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17 (Springer, Berlin, 1991).Google Scholar
Monod, N., Arithmeticity vs. nonlinearity for irreducible lattices , Geom. Dedicata 112 (2005), 225–237.Google Scholar
Monod, N., Superrigidity for irreducible lattices and geometric splitting , J. Amer. Math. Soc. 19 (2006), 781–814.Google Scholar
Raghunathan, M. S., Discrete subgroups of algebraic groups over local fields of positive characteristic , Proc. Indian Acad. Sci 99 (1989), 127–146.Google Scholar
Rosendal, C., Automatic continuity of group homomorphisms , Bull. Symb. Log. 15 (2009), 184–214.Google Scholar
Serre, J.-P., Lie algebras and Lie groups, Lecture Notes in Mathematics, vol. 1500 (Springer, Berlin, 2006), 1964 lectures given at Harvard University, corrected fifth printing of the second (1992) edition.Google Scholar
Tits, J., Free subgroups in linear groups , J. Algebra 20 (1972), 250–270; MR 0286898.Google Scholar
Zimmer, R. J., Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81 (Birkhäuser, Basel, 1984); MR 776417 (86j:22014).Google Scholar