Abstract: We shall discuss ergodic properties of the action of a subgroup H of a free group F on the Poisson boundary of the simple random walk on F. Methods from combinatorial group theory will be used to identify the conservative and the dissipative part of the action. We also present necessary and sufficient conditions of conservativity of the action in terms of geometry of the quotient. This is a joint work with R. Grigorchuk and V. Kaimanovich. |
Abstract: We discuss rigidity and classification results for Borel actions of countable groups on quotients of Polish spaces by countable Borel equivalence relations. |
Abstract : We shall discuss when a four-manifold admits a pair of complementary two-dimensional foliations. After some generalities and definitions, we discuss in detail two specific situations when a lot more than in the general case can be said for symplectic four-manifolds: pairs of complementary Lagrangian foliations (in Lecture 1), and pairs of complementary symplectic foliations (in Lecture 2). The two discussions are independent of each other. Both have relations to dynamical questions, about Anosov systems in the first case, and about symplectomorphism groups in the second case. |
Abstract : In joint work with Kai-Uwe Bux, we show how techniques in the geometry of arithmetic groups over number fields provide negative results about the finiteness properties of arithmetic groups over function fields. |
Abstract : Ramanujan graphs are k-regular graphs with optimal bounds on their eigenvalues. They play a central role in various questions in combinatorics and computer science. Their construction is based on the work of Deligne and Drinfeld on the Ramanujan conjecture for GL(2). The recent work of Lafforge which settles the Ramanujan conjecture for GL(n) over function fields opens the door to the study of Ramanujan complexes: these are higher dimensional analogues which are obtained as quotients of the Bruhat-Tits building of PGL(n) over local fields. |
Abstract : We will discuss a new method for obtaining identities for the spectrum of lengths of simple geodesics on a hyperbolic surface with at least one boundary component. Our method is based on : 1 : Wolpert's formula for the variation of geodesic lengths 2 : the polynomial growth rate of the aforementioned spectrum 3 : the existence of an element of finite order in the mapping class group. We will discuss the advantages of this method over the original method (McShane's thesis) and Bowditch's Markoff maps. |
Abstract : We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental group, namely they are precisely the Moebius band, and the products of R with the total spaces of flat Euclidean vector bundles over compact infranilmanifolds. The proof uses collapsing theory. |
Abstract :These talks are aimed at non-experts (so primarily students). The goal is to explain the background needed to understand, for example, topological results on mapping class groups proved using Hodge theory. |
Abstract : We shall discuss measurable rigidity phenomena for group actions on infinite homogeneous spaces, such as the following result: THM: Suppose that two abstractly isomorphic lattices L_1 and L_2 in SL(2,R) admit a measurable map T of the plane R^2 which intertwines their linear actions. Then L_1 and L_2 are necessarily conjugate and T is a linear map realizing this conjugation. This theorem is due to Y.Shalom and T.Steger, who deduce this and similar rigidity results from the study of unitary representations of lattices. We shall discuss a purely dynamical approach which gives a broader picture of measurable rigidity on certain infinite measure spaces. Curiously, the above theorem can be considered as a dual to the horocycle rigidity results of M. Ratner (1982). In this case this duality is fruitful in posing questions but does not seem to help in proofs. |
Abstract: I will talk about a theorem about the stationarity of Patterson-Sullivan measure on the boundary of CAT(-1) spaces. I will explain how to build this measure and how to insure the first moment condition provided the underlying group action is co-compact. I will also sketch the proof by Furstenberg showing stationarity of Haar measure on the boundary G/P, where G is a semisimple Lie group and P is a minimal parabolic. |
Abstract: Outline: I. Locally homogeneous geometric structures in the sense of Ehresmann II. Relation to representations of the fundamental group. Developing maps and holonomy representations. III. Pathological versus tame developing maps; Real and Complex Projective structures IV. Generalized Fenchel-Nielsen coordinates V. Dynamics on the SU(2)-moduli space VI. Spaces of Real Characters of Rank Two free groups |
Abstract: Outline: I. Locally homogeneous geometric structures in the sense of Ehresmann II. Relation to representations of the fundamental group. Developing maps and holonomy representations. III. Pathological versus tame developing maps; Real and Complex Projective structures IV. Generalized Fenchel-Nielsen coordinates V. Dynamics on the SU(2)-moduli space VI. Spaces of Real Characters of Rank Two free groups |
Abstract: This is a joint work with Tsachik Gelander. A group action on a set is called primitive if there is no non trivial invariant equivalence relation on the set. A group is primitive if it admits a faithful primitive action. Our goal is to understand which finitely generated groups are primitive. We will give a complete answer to this question for linear groups, and for subgroups of hyperbolic groups. We will also give partial answers for solvable groups and for subgroups of mapping class groups. Our methods come form earlier works due to Margulis and Soifer; Breuillard and Gelander; and from an earlier joint work with Miklos Abert. |