Orateur: Michelle Bucher-Karlsson  (Unige):
Titre: Volume and characteristic numbers of representations of hyperbolic manifolds.
Résumé:  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 rigidity properties have been studied extensively. We will show how to define the volume of a representation in the noncompact case (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 value for the volume of a representation is discrete. In even dimension, this follows from the fact tha the volume 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 3, the set of value of the volume of a representation is, up to a universal constant, an integer.
Restricting to 3-dimensional manifolds, we can further study the representations of hyperbolic lattices into SL(m,C) and we will show that the maximal representation is the geometric geometric representation, i.e the composition of the lattice embedding G --> SL(2,C) with the irreducible representation SL(2,C) --> SL(n,C). (This is joint work with Marc Burger and Alessandra Iozzi.)

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