Talks:
11:00 - 12:00 am (Room 109): Elon Lindenstrauss, Princeton University
Periodic Orbits on the Space of Lattices
Abstract:
In 1967 Furstenberg proved that for any irrational x,
the set
A related survey paper Rigidity of Multiparameter Actions is here.
1:00 - 2:00 pm (Room 1302): David Fisher, Indiana University - Bloomington
Coarse differentiation of quasi-isometries and rigidity for lattices in solvable Lie groups.
Abstract: In the early 80's Gromov initiated a program to study finitely generated groups up to quasi-isometry. This program was motivated in part by rigidity properties of lattices in Lie groups. A lattice &Gamma in a group G is a discrete subgroup where the quotient G / &Gamma has finite volume. A major theorem of Gromov's in this direction is a rigidity result for lattices in nilpotent Lie groups.
In the 1990's, a series of dramatic results led to the completion of the Gromov program for lattices in semisimple Lie groups. The next natural class of examples to consider are lattices in solvable Lie groups, and even results for the simplest examples were elusive for a considerable time. I will discuss joint work with Eskin and Whyte in which we prove the first results on quasi-isometry classification of lattices in non-nilpotent, solvable Lie groups. In particular, we prove that any group quasi-isometric to the three dimensional solvable group Sol is virtually a lattice in Sol. The results are proven by a method of coarse differentiation, which I will outline.
4:00 - 5:00 pm (Room 1302): Anna Wienhard, Institute for Advanced Study
Bounded cohomology and applications to rigidity
Abstract: Bounded cohomology provides tools to study group homomorphisms of a finitely generated group &Gamma into a semisimple Lie group G.
When G ia a semisimple Lie group of Hermitian type we can associate to a homomorphism &rho: &Gamma &rarr G an invariant with values in the second (continuous) bounded cohomology of &Gamma. This invariant, called the bounded Kahler class of &rho allows to study different classes of homomorphisms as well as the moduli space Hom(\G,G)/G.
If for example if G = SU(p,q), with p not equal to q, then homomorphisms with a Zariski dense image are, up to equivalence, completely determined by this bounded Kahler class.
I will discuss this and further applications of the bounded Kahler class to study group homomorphisms into Lie groups of Hermitian type.
This is joint work with M. Burger and A. Iozzi
5:30 -6:30 pm (Room 1302): Tsachik Gelander, Yale University
Super-rigidity, generalized harmonic maps, and uniformly convex metric spaces.
Abstract: I will speak about a joint work with A. Karlsson and G.A. Margulis.
Since Margulis proved his remarkable superrigidity theorem, various extensions and generalizations of it were proved by various people.
The superrigidity theorem can be read as follows: Let G be a locally compact group and L a lattice in G. Let X be a space on which L acts by isometries. Then, under an appropriate assumptions on G,L,X and the action, the action extends continuously to G. In Margulis' classical theorem G is a higher rank semisimple Lie group, L an irreducible lattice, X a symmetric space of non-compact type and the action is unbounded and Zariski dense (in Isom(X)).
We proved a superrigidity theorem for irreducible lattices L in a product of general locally compact groups G=G1x...xGn (n>1), where X is a general metric space whose metric satisfies some convexity assumptions.
This generalize a recent theorem of N. Monod who establish the same result when X is a CAT(0) space.
Our proof uses the notion of generalized harmonic maps, and is influenced by a preprint of Margulis (from ~ 1990) about superrigidity for commensurable groups.
Organizers:
No registration is necessary. Bring ID to show the guard.