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Tim de Laat; Masato Mimura; Mikael de la Salle
On strong property (T) and fixed point properties for Lie groups
(Sur la propriété (T) renforcée et la propriété de point fixe pour les groupes de Lie)
Annales de l'institut Fourier, 66 no. 5 (2016), p. 1859-1893, doi: 10.5802/aif.3051
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Class. Math.: 20J06, 22D12, 22E45, 46B20
Keywords: Strong property (T), Banach space representations, geometry of Banach spaces, bounded cohomology

Résumé - Abstract

We consider certain strengthenings of property (T) relative to Banach spaces. Let $X$ be a Banach space for which the Banach–Mazur distance to a Hilbert space of all $k$-dimensional subspaces grows as a power of $k$ strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank has strong property (T) of Lafforgue with respect to $X$. As a consequence, every continuous affine isometric action of such a high rank group (or a lattice in such a group) on $X$ has a fixed point. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. We prove that every special linear group of sufficiently large rank satisfies the following property: every quasi-$1$-cocycle with values in an isometric representation on $X$ is bounded.


[1] Uri Bader, Alex Furman, Tsachik Gelander & Nicolas Monod, “Property (T) and rigidity for actions on Banach spaces”, Acta Math. 198 (2007) no. 1, p. 57-105 Article |  Zbl 1162.22005
[2] Uri Bader, Christian Rosendal & Roman Sauer, “On the cohomology of weakly almost periodic group representations”, J. Topol. Anal. 6 (2014) no. 2, p. 153-165 Article |  Zbl 1298.22005
[3] Bachir Bekka, Pierre de la Harpe & Alain Valette, Kazhdan’s property (T), New Mathematical Monographs 11, Cambridge University Press, Cambridge, 2008 Article |  Zbl 1146.22009
[4] M. Burger & N. Monod, “Continuous bounded cohomology and applications to rigidity theory”, Geom. Funct. Anal. 12 (2002) no. 2, p. 219-280 Article |  Zbl 1006.22010
[5] David Carter & Gordon Keller, “Bounded elementary generation of ${\rm SL}_{n}(\mathcal{O})$”, Amer. J. Math. 105 (1983) no. 3, p. 673-687 Article |  Zbl 0525.20029
[6] Cordelia Druţu & Piotr W. Nowak, “Kazhdan projections, random walks and ergodic theorems”, http://arxiv.org/abs/1501.03473, 2015
[7] E. B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras”, Mat. Sbornik N.S. 30(72) (1952), p. 349-462 (3 plates)  Zbl 0048.01701
[8] E. B. Dynkin, Selected papers of E. B. Dynkin with commentary, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000, Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik  Zbl 1056.01014
[9] David B. A. Epstein & Koji Fujiwara, “The second bounded cohomology of word-hyperbolic groups”, Topology 36 (1997) no. 6, p. 1275-1289 Article |  Zbl 0884.55005
[10] Mikhail Ershov & Andrei Jaikin-Zapirain, “Property (T) for noncommutative universal lattices”, Invent. Math. 179 (2010) no. 2, p. 303-347 Article |  Zbl 1205.22003
[11] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978  Zbl 0993.53002
[12] Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, p. 187–204  Zbl 0034.10503
[13] D. A. Každan, “On the connection of the dual space of a group with the structure of its closed subgroups”, Funkcional. Anal. i Priložen. 1 (1967), p. 71-74  Zbl 0168.27602
[14] Tim de Laat & Mikael de la Salle, “Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders”, to appear in J. Reine Angew. Math., http://arxiv.org/abs/1403.6415, 2015
[15] Tim de Laat & Mikael de la Salle, “Strong property (T) for higher-rank simple Lie groups”, Proc. Lond. Math. Soc. (3) 111 (2015) no. 4, p. 936-966 Article
[16] Vincent Lafforgue, “Un renforcement de la propriété (T)”, Duke Math. J. 143 (2008) no. 3, p. 559-602 Article |  Zbl 1158.46049
[17] Vincent Lafforgue, “Propriété (T) renforcée banachique et transformation de Fourier rapide”, J. Topol. Anal. 1 (2009) no. 3, p. 191-206 Article |  Zbl 1186.46022
[18] Vincent Lafforgue & Mikael de la Salle, “Noncommutative $L^p$-spaces without the completely bounded approximation property”, Duke Math. J. 160 (2011) no. 1, p. 71-116 Article |  Zbl 1267.46072
[19] Benben Liao, “Strong Banach property (T) for simple algebraic groups of higher rank”, J. Topol. Anal. 6 (2014) no. 1, p. 75-105 Article |  Zbl 1291.22010
[20] Masato Mimura, “Fixed point properties and second bounded cohomology of universal lattices on Banach spaces”, J. Reine Angew. Math. 653 (2011), p. 115-134 Article |  Zbl 1221.20032
[21] Masato Mimura, “Strong algebraization of fixed point properties”, http://arxiv.org/abs/1505.06728, 2015
[22] Masato Mimura & Hiroki Sako, “Group approximation in Cayley topology and coarse geometry, part II: Fibered coarse embeddings”, in preparation
[23] Igor Mineyev, Nicolas Monod & Yehuda Shalom, “Ideal bicombings for hyperbolic groups and applications”, Topology 43 (2004) no. 6, p. 1319-1344 Article |  Zbl 1137.20033
[24] Nicolas Monod, Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics 1758, Springer-Verlag, Berlin, 2001 Article |  Zbl 0967.22006
[25] Nicolas Monod, An invitation to bounded cohomology, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, p. 1183–1211  Zbl 1127.55002
[26] Nicolas Monod & Yehuda Shalom, “Cocycle superrigidity and bounded cohomology for negatively curved spaces”, J. Differential Geom. 67 (2004) no. 3, p. 395-455  Zbl 1127.53035
[27] Izhar Oppenheim, “Averaged projections, angles between groups and strengthening of property (T)”, http://arxiv.org/abs/1507.08695, 2015
[28] Gilles Pisier & Quan Hua Xu, Random series in the real interpolation spaces between the spaces $v_p$, Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math. 1267, Springer, Berlin, 1987, p. 185–209 Article |  Zbl 0634.46009
[29] Mikael de la Salle, “Towards Strong Banach property (T) for $\mathrm{SL}(3,\mathbb{R})$”, to appear in Israel J. Math., http://arxiv.org/abs/1307.2475, 2015
[30] Yehuda Shalom, “Bounded generation and Kazhdan’s property (T)”, Inst. Hautes Études Sci. Publ. Math. (1999) no. 90, p. 145-168 (2001) Numdam |  Zbl 0980.22017
[31] Nicole Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics 38, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989  Zbl 0721.46004
[32] William A. Veech, “Weakly almost periodic functions on semisimple Lie groups”, Monatsh. Math. 88 (1979) no. 1, p. 55-68 Article |  Zbl 0438.43009