| Abstract: In joint work with G.A.Margulis, we have proven many results
showing that smooth actions of "large groups" on compact manifolds are locally rigid.
By local rigidity we mean that any perturbation of the action is conjugate back
to the original action by a diffeomorphism. The exact meaning of this depends on
which topology ($C^k, C^{\infty}$) one uses to define perturbation and the smoothness of
the resulting conjugacy. We have recently dramatically improved our results
in this direction and can now prove many of our results for perturbations that are
only $C^2$ or $C^3$ close to the original action and also produce $C^{\infty}$ conjugacies
as long as the perturbation is $C^{\infty}$ action. In order to be able to discuss
some of the ideas in the proofs and some relations to KAM theory, after a brief introduction, I
will concentrate on the following:
Theorem. Let G be a group with property T of Kazhdan. Then any isometric action of G on a compact Riemannian manifold is locally rigid. |
| Abstract: Consider the space of $C^r$ diffeomorphisms (smooth invertible selfmaps) of a compact surface $M$ (e.g. $S^2$ or $T^2$) Diff$^r(M)$ with $r\geq 2$. A sink of $f:M \to M$ is a periodic point $x \in M$ which attract all points from its neighbourhood (as in your kitchen). Points attracted to $x$ called basin of attraction of $x$. In 60-th Thom conjectured that a generic diffeomorphism can not have infinitely many coexisting sinks. Indeed, each sink has an open basin of attraction and infinitely many of those seems too much. In 70-th Newhouse constructed an open set of diffeomophisms $N \subset \textup{Diff}^r(M)$ such that generic diffeomorphism in $N$ does have infinitely many coexisting sinks. It is an amazing phenomenon, called Newhouse phenomenon. It disproves Thom's conjecture and significant obstacle to describe ergodic properties of surface diffeomorphisms. We shall discuss this phenomenon and closely related results of Benedicks-Carleson, Mora-Viana, Wang-Young, Morreira-Yoccoz. The main result indicates in some sense this phenomenon has "probability zero". This is a particular case of so-called Palis conjecture. This is a joint work with A. Gorodetsky |
| Abstract: I will describe a joint work with T. Gelander. In his celebrated 1972 paper, J. Tits proved the following dichotomy for linear groups: Tits Alternative: Let K be a field and G a subgroup of GL(n,K), in case char(K) > 0 assume further that G is finitely generated. Then either G contains a solvable subgroup of finite index, or G contains a non commutative free subgroup. We proved a topological version of this theorem. Suppose k is a local field, that is R, C, a finite extension of a p adic field or a field of formal power series in one variable over a finite field. Then the topology of k induces a topology on GL(n,k) and on any subgroup of it. Theorem: Let G be any subgroup of GL(n,k). Then either G contains a relatively open solvable subgroup or G contains a relatively dense free subgroup. For example, the group SL(n,Q) contains free subgroup of rank 2 which is dense with respect to the real topology, and similarly for any prime p it contains a dense free subgroup of rank r(p) which is dense with respect to the p-adic topology. For k=R our theorem answers a question of Carriere and Ghys and provides a short proof for a conjecture of Connes and Sullivan on amenable actions which was first proved by Zimmer by means of super rigidity methods. For k non Archimedean our theorem implies a conjecture of Dixon, Pyber, Seress and Shalev on profinite groups, as well as a conjecture of Shalev on coset identity in pro p groups (in the analytic case). We also apply our theorem to prove a conjecture of Carriere, that the volume growth of the leaves in any Riemannian foliation on a compact manifold is either polynomial or exponential. |
| Abstract: In these talks I will explain how the structure of mapping class and Torelli groups leads to previously unknown expressions in cohomology for classes of certain loci in Deligne-Mumford moduli spaces in terms of tautological classes. |
| Abstract: In these talks I will explain how the structure of mapping class and Torelli groups leads to previously unknown expressions in cohomology for classes of certain loci in Deligne-Mumford moduli spaces in terms of tautological classes. |