On strong property (T) and fixed point properties for Lie groups

de Laat, Tim; Mimura, Masato; de la Salle, Mikael

Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1859-1893.

Nous considérons certains renforcements de la propriété (T) Banachique. Soit $X$ un espace de Banach pour lequel la distance de Banach–Mazur à un espace euclidien de tout sous-espace de dimension $k$ croît comme une puissance de $k$ strictement inférieure à un demi. Nous prouvons que tout groupe de Lie simple connexe et de rang réel suffisament grand a la propriété (T) renforcée de Lafforgue relativement à $X$. Par conséquent toute action continue par isométries affines d’un tel groupe (ou d’un réseau dans un tel groupe) sur $X$ a un point fixe. Pour les groupes spéciaux linéaires, nous présentons aussi une approche plus directe aux propriétés de point fixe. Plus précisément nous prouvons que tout groupe spécial linéaire de rang suffisament grand a la propriété suivante : tous ses quasi-$1$-cocycles à valeurs dans une représentations isométrique sur $X$ sont bornés.

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.

Classification : 20J06, 22D12, 22E45, 46B20
Keywords: Strong property (T), Banach space representations, geometry of Banach spaces, bounded cohomology
Mot clés : Propriété (T) renforcée, représentations sur des espaces de Banach, géométrie des espaces de Banach, cohomologie bornée

[1] Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas Property (T) and rigidity for actions on Banach spaces, Acta Math., Volume 198 (2007) no. 1, pp. 57-105 | DOI

[2] Bader, Uri; Rosendal, Christian; Sauer, Roman On the cohomology of weakly almost periodic group representations, J. Topol. Anal., Volume 6 (2014) no. 2, pp. 153-165 | DOI

[3] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008, xiv+472 pages | DOI

[4] Burger, M.; Monod, N. Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 219-280 | DOI

[5] Carter, David; Keller, Gordon Bounded elementary generation of ${\mathrm{SL}}_n\left(𝒪\right)$, Amer. J. Math., Volume 105 (1983) no. 3, pp. 673-687 | DOI

[6] Druţu, Cordelia; Nowak, Piotr W. Kazhdan projections, random walks and ergodic theorems (2015) (http://arxiv.org/abs/1501.03473)

[7] Dynkin, E. B. Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S., Volume 30(72) (1952), p. 349-462 (3 plates)

[8] Dynkin, E. B. Selected papers of E. B. Dynkin with commentary, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000, xxviii+796 pages (Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik)

[9] Epstein, David B. A.; Fujiwara, Koji The second bounded cohomology of word-hyperbolic groups, Topology, Volume 36 (1997) no. 6, pp. 1275-1289 | DOI

[10] Ershov, Mikhail; Jaikin-Zapirain, Andrei Property (T) for noncommutative universal lattices, Invent. Math., Volume 179 (2010) no. 2, pp. 303-347 | DOI

[11] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978, xv+628 pages

[12] John, Fritz 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, pp. 187-204

[13] Každan, D. A. On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen., Volume 1 (1967), pp. 71-74

[14] de Laat, Tim; de la Salle, Mikael Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders (2015) (to appear in J. Reine Angew. Math., http://arxiv.org/abs/1403.6415)

[15] de Laat, Tim; de la Salle, Mikael Strong property (T) for higher-rank simple Lie groups, Proc. Lond. Math. Soc. (3), Volume 111 (2015) no. 4, pp. 936-966 | DOI

[16] Lafforgue, Vincent Un renforcement de la propriété (T), Duke Math. J., Volume 143 (2008) no. 3, pp. 559-602 | DOI

[17] Lafforgue, Vincent Propriété (T) renforcée banachique et transformation de Fourier rapide, J. Topol. Anal., Volume 1 (2009) no. 3, pp. 191-206 | DOI

[18] Lafforgue, Vincent; de la Salle, Mikael Noncommutative $L^p$-spaces without the completely bounded approximation property, Duke Math. J., Volume 160 (2011) no. 1, pp. 71-116 | DOI

[19] Liao, Benben Strong Banach property (T) for simple algebraic groups of higher rank, J. Topol. Anal., Volume 6 (2014) no. 1, pp. 75-105 | DOI

[20] Mimura, Masato Fixed point properties and second bounded cohomology of universal lattices on Banach spaces, J. Reine Angew. Math., Volume 653 (2011), pp. 115-134 | DOI

[21] Mimura, Masato Strong algebraization of fixed point properties (2015) (http://arxiv.org/abs/1505.06728)

[22] Mimura, Masato; Sako, Hiroki Group approximation in Cayley topology and coarse geometry, part II: Fibered coarse embeddings (in preparation)

[23] Mineyev, Igor; Monod, Nicolas; Shalom, Yehuda Ideal bicombings for hyperbolic groups and applications, Topology, Volume 43 (2004) no. 6, pp. 1319-1344 | DOI

[24] Monod, Nicolas Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, 1758, Springer-Verlag, Berlin, 2001, x+214 pages | DOI

[25] Monod, Nicolas An invitation to bounded cohomology, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1183-1211

[26] Monod, Nicolas; Shalom, Yehuda Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom., Volume 67 (2004) no. 3, pp. 395-455 http://projecteuclid.org/euclid.jdg/1102091355

[27] Oppenheim, Izhar Averaged projections, angles between groups and strengthening of property (T) (2015) (http://arxiv.org/abs/1507.08695)

[28] Pisier, Gilles; Xu, Quan Hua Random series in the real interpolation spaces between the spaces $v_p$, Geometrical aspects of functional analysis (1985/86) (Lecture Notes in Math.), Volume 1267, Springer, Berlin, 1987, pp. 185-209 | DOI

[29] de la Salle, Mikael Towards Strong Banach property (T) for $\mathrm{SL}(3,ℝ)$ (2015) (to appear in Israel J. Math., http://arxiv.org/abs/1307.2475)

[30] Shalom, Yehuda Bounded generation and Kazhdan’s property (T), Inst. Hautes Études Sci. Publ. Math. (1999) no. 90, p. 145-168 (2001) | DOI

[31] Tomczak-Jaegermann, Nicole 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, xii+395 pages

[32] Veech, William A. Weakly almost periodic functions on semisimple Lie groups, Monatsh. Math., Volume 88 (1979) no. 1, pp. 55-68 | DOI