On Property (T) for Discrete Groups

We present a simple sufficient condition which enables one to prove property (T) for a discrete group from its presentation and to compute the Kazhdan constants. This condition applies to some lattices for which property (T) was known and gives a new elementary proof. Using this condition one can construct new examples of Kazhdan groups and finally prove that random groups in the sense of Gromov are infinite, hyperbolic and have property (T).

[1]  Robert J. Zimmer,et al.  Ergodic Theory and Semisimple Groups , 1984 .

[2]  Sylvain Barré.  Sur les polyèdres de rang 2 , 1997 .

[3]  Béla Bollobás,et al.  Random Graphs , 1985 .

[4]  P. Pansu.  Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles , 1998 .

[5]  Gregory Margulis,et al.  Finitely-additive invariant measures on Euclidean spaces , 1982, Ergodic Theory and Dynamical Systems.

[6]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[7]  Walter Feit,et al.  The nonexistence of certain generalized polygons , 1964 .

[8]  A. Zuk,et al.  La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres , 1996 .

[9]  Dennis Sullivan,et al.  For $n > 3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets , 1981 .

[10]  T. Gelander,et al.  Dependence of Kazhdan constants on generating subsets , 2002 .

[11]  Howard Garland,et al.  p-Adic Curvature and the Cohomology of Discrete Subgroups of p-Adic Groups , 1973 .

[12]  G. Skandalis.  Une Notion de nucléarité en K-théorie (d'après J. Cuntz) , 1988 .

[13]  D. I. Cartwright,et al.  Property (T) and $\overline A_2$ groups , 1994 .

[14]  W. Ballmann,et al.  On L2-cohomology and Property (T) for Automorphism Groups of Polyhedral Cell Complexes , 1997 .

[15]  J. W. S. Cassels,et al.  Geometric Group Theory: Asymptotic Invariants of Infinite Groups, M. Gromov , 1993 .

[16]  Claire Delaroche,et al.  Sur les relations entre l'espace dual d'un groupe et la structure de ses sous-groupes fermés , 1968 .

[17]  Pierre de la Harpe,et al.  La propriété (T) de Kazhdan pour les groupes localement compacts , 1989 .

[18]  D. Kazhdan.  Connection of the dual space of a group with the structure of its close subgroups , 1967 .

[19]  Christophe Champetier.  Propriétés génériques des groupes de type fini , 1991 .

[20]  Andrzej Zuk,et al.  Property (T) and Kazhdan constants for discrete groups , 2003 .

[21]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[22]  P. Pansu.  Sous-groupes discrets des groupes de Lie : rigidité, arithméticité , 1994 .

[23]  V. Drinfel'd,et al.  Finitely additive measures on S2 and S3, invariant with respect to rotations , 1984 .