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Group Actions

Fall 2009

Tuesday 12:30-1:30 pm
Fine 322

Lunch will be provided!

Oct 27 Anna Wienhard
Princeton University
Domains of Discontinuity with compact quotients

Abstract: Given a discrete subgroup in a semisimple Lie group G, it is natural to ask on which homogeneous spaces this subgroup acts properly discontinuous and with compact quotient. In this talk I will discuss a large class of examples of such subgroups and explain a construction of open subsets in G/P where P is a parabolic subgroup, on which these groups act properly discontinuous with compact quotient. The class of examples I am going to discuss includes subgroups arising from higher Teichmueller spaces, and finding geometric structures parametrized by higher Teichmueller spaces is an important motivation for this work. This is joint work with O. Guichard.
Nov. 10 Nimish Shah
Ohio State University
Equidistribution and counting points on orbits of geometrically finite hyperbolic groups

Abstract: In this joint work with Hee Oh, we consider various sphere packing configurations that are invariant under actions of geometrically finite hyperbolic groups, and estimate the cardinality of spheres of curvature at most T with respect to euclidean, or spherical, or hyperbolic metric. This sphere counting problem is studied by proving "weighted equidistribution" results related to translates of certain co-dimension one submanifolds under the geodesic flow on the unit tangent bundle of M a hyperbolic n-manifold, where the fundamental group of M is a geometrically finite discrete subgroup of the group of isometries of the n dimensional hyperbolic space.
Nov. 17 Amir Mohammadi
University of Chicago
Inhomogeneous quadratic forms

Abstract: We will address a recent join work with G. Margulis on a quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. This generalizes the previous works of Eskin, Margulis and Mozes in the homogeneous setting also the work of J. Marklof.
Nov. 24 Zhiren Wang
Princeton University
Effective results on actions of commuting toral automorphisms

Abstract: Let G be an abelian subgroup of SL(d,Z). When G acts totally irreducibly on T^d the d-dimensional torus, has some hyperbolicity and is not virtually-cyclic, Berend proved that every orbit on T^d is either the whole torus or finite. We will discuss effective forms of this theorem and how they are related to number-theoretical problems. This is an analogue of the recent quantitative Furstenberg's theorem concerning the X 2, X 3 action (times 2, times 3 action) on the circle by Bourgain-Lindenstrauss-Michel-Venkatesh.
Dec. 1 Martin Kassabov
Cornell University/IAS
Subspace arrangements and property T

Abstract: will talk about my viewpoint at a method for proving property T developed by Dymara and Januszkiewicz. Their original motivation came from groups acting on buildings, but the idea does not used anything more angles between subspaces in an (finite dimensional) Euclidian space. The main result says that if a group G is generated by finite subgroups G_i and each pair generates a group with property T and sufficiently large Kazhdan constant then the whole group also has property T. One can use this result to show that some groups like SL_n(F_p[t_1,...,t_k]) has property T, almost without using any representation theory. Another application allow us to compute the exact values of the Kazhdan constant and the spectral gap for the Laplacian for any finite Coxeter group with respect to its standard generating set.
Dec. 8 Uri Shapira
Hebrew University
Homogeneous orbit closures and Diophantine approximations of algebraic numbers.

Abstract: The content of the talk is a joint work with Elon Lindenstrauss. Let X be the space of unimodular (covolume 1) lattices in Euclidean d-space and let A denote the group of diagonal matrices of determinant 1. We prove that any lattice x in X which "comes from a number field" which is not a CM field satisfies a Ratner-like property, namely the closure of the orbit Ax equals to an orbit Hx of a group H containing A. As a consequence I generalize my previous work on Diophantine properties of totally real cubic numbers by droping the dimension assumption and the totally realness.
Dec. 11
3:30 pm

Same room
Fine 322
Peter Varju
Princeton University
Expansion in SL(d, Z/qZ), q square-free.

Abstract: I discuss the problem whether certain Cayley graphs form an expander family. A family of graphs is called an expander family, if the number of edges needed to be deleted from any of the graphs to make it disconnected is at least a constant multiple of the size of the smallest component we get. Let S be a subset of SL(d, Z) closed for taking inverses. For each square-free integer q consider the graph whose vertex-set is SL(d, Z/qZ) two of which is connected by an edge precisely if we can get one from the other by left multiplication by an element of S. Bourgain, Gamburd and Sarnak proves that if d = 2 and S generates a Zariski dense subgroup of SL2, then these graphs form an expander family. In the talk I outline a modification of their argument which leads to a simpler proof and allows a generalization to d = 3 or to general number fields.

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