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Group Actions Seminar

Spring 2008

Monday 12:15-1:15 pm
Fine Hall 224

February 11
No Meeting
February 18 Alex Gorodnik
University of Bristol/Princeton University
Regularity of conjugacy for actions of large groups
It is well know that a small perturbation of an Anosov map is topologically conjugate to the original map, but the conjugacy is not smooth in general. We prove that for actions of "large" (e.g., Zariski dense) groups topological conjugacy is smooth. This is a joint work with Hitchman and Spatzier.
February 25
No Meeting
March 3 Amir Mohammadi
Yale University
Unipotent flows in positive characteristic
Study of dynamics of action of unipotent subgroups on homogeneous spaces and its applications to Diophantine approximations has been attracting considerable attention over the past 40 years or so. Margulis's celebrated proof of Oppenheim conjecture and Ratner's seminal work on Raghunathan's conjecture are two inspiring works in the subject. Although Raghunathan's conjectures in characteristic zero have been settled affirmatively, very little is known in positive characteristic case. In this talk we will address this issue. In particular the main focus will be on a recent joint work with M. Einsiedler on classification of joinings for the action of certain unipotent subgroups. As it turns out this has some applications to quasi-isometries of lattices. This connection is drawn explicit by K. Wortman in an appendix to our work.
March 10 Alireza Salehi Golsefidy
Princeton University
Translates of horospherical measures and counting problems.
(Joint with A. Mohammadi) In this talk, I will briefly explain the relation between some of the counting problems, mixing, and ergodic theory. The counting problems might be of geometric or number theoretic nature. For instance consider V=G/H a homogeneous variety, and one would like to study the integer or rational points on V. Eskin, Mozes, and Shah attacked this problem via unipotents flows. However they had to assume that H is maximal and reductive (in particular not inside any parabolic subgroup of G.) I will explain an ergodic theoretic approach toward such problem for a flag variety.

For a geometric example, consider SL(n,Z)-translates of a horosphere in the symmetric space of SL(n,R). Question is how many of them intersect a ball of radius R. In fact, Eskin and McMullen answered this question for n=2, using mixing. I will explain why mixing is not enough and how one can get such a result for any n.

I will show that the main ingredient for both of the mentioned questions is understanding the limits of translates of horospherical measures, i.e. the probability measure supported on U SL(n,Z)/SL(n,Z), where U is the set of upper triangular unipotent matrices.
March 24
No meeting
March 31 Danijela Damjanovic
Harvard University
Local rigidity for some rank two algebraic abelian actions
I will discuss local rigidity for some rank-two actions: partially hyperbolic on SL(n, R)/L, for n>3; and parabolic on SL(2, R)XSL(2, R)/L.
April 7 Kariane Calta
Vassar College
Translation Surfaces and Notions of Periodicity
I will provide a brief introduction to translation surfaces, which are surfaces that can be constructed by gluing finitely many polygons in R^2 along parallel edges to form a closed surface with cone points. I will discuss the geometric notion of complete periodicity as it relates to the classification of lattice translation surfaces in genus two given by Calta and McMullen. Then I will introduce the notion of algebraic periodicity which generalizes that of complete periodicity and discuss recent results of Calta and Smillie related to algebraic periodicity.
April 14 Speaker
No meeting
April 21 Francois Maucourant
Rennes 1 University
Two examples of nonhomogeneous orbit closures.
We will explain how to construct orbits of non-homogeneous closure for some subgroup of the diagonal group acting on the space of lattices of dimension at least 6. Also, a similar example for the action of *2,*3 on the four dimensionnal torus is discussed.
April 28 Alex Furman
University of Illinois at Chicago
L^1-Measure Equivalence, Simplicial volume, and rigidity of hyperbolic lattices.
I will report on a recent work with Uri Bader and Roman Sauer on Orbit Equivalence (or Measure Equivalence) of lattices in SO(n,1), n>2. Measure Equivalence (ME) is a rather weak equivalence relation between groups, which may be viewed as a measurable analogue of Quasi-Isometry. This concept is closely related to questions of orbit structures of group actions in Ergodic Theory, and has connections to von Neumann algebras and Descriptive set theory. Most existing rigidity results are related to higher rank phenomena, while most ME-invariants of groups (such as amenability, property (T), L^2-Betti numbers) involve unitary representations. In this work we consider a restricted notion of ME, which seems to be prevalent among actions of non-amenable groups. In this context we prove rigidity results for all lattices in SO(n,1), and show that in the wider class of fundamental groups of negatively curved manifolds, the dimension and the simplicial volume (a topological invariant) are also preserved by L^1-OE.

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