Cari tutti,

vi ricordo che questa settimana, oltre al seminario di A.G. Lecuona che si svolgera' mercoledi' alle 14.30, vi sara' un seminario di geometria anche domani alle 15, sempre in aula seminari. Il seminario avra' l'usuale durata di un'ora, e si svolgera' percio' dalle 15 alle 16 (la durata di un'ora e mezza segnalata dal settimanale e' errata). Eccone i dettagli:

Isometric properties of bounded cohomology and applications to volumes of representations

Michelle Bucher (Università di Ginevra)

Let G be a lattice in SO(n,1) and let h:G --> SO(n,1) be any representation. For cocompact lattices, the volume of a representation is an invariant whose maximal and rigidity properties have been studied extensively. We will show how the fact that the isomorphism between the bounded cohomology of a hyperbolic manifold and the relative bounded cohomology of the manifold relative its boundary is isometric can be used to define the volume of a representation (a different definition by Francaviglia and Klaff also exists). In particular, we establish a rigidity result for maximal representations, recovering Mostow rigidity for hyperbolic manifolds. In the cocompact case, the set of values for the volume of a representation is discrete. In even dimension, this follows from the fact that the volume form is an Euler class. In odd dimension, this was proven by Besson, Courtois and Gallot. The situation changes in the noncocompact case and for example the discreteness of the set of value is not valid anymore in dimension 2 and 3. We prove that in even dimension greater or equal to 4, the set of value of the volume of a representation is, up to a universal constant, an integer. This is joint work with Marc Burger and Alessandra Iozzi.