of Discontinuity with compact quotients
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.
and counting points on orbits of
geometrically finite hyperbolic groups
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
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
results on actions of commuting toral automorphisms
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
|Subspace arrangements and property T
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
The main result says that if a group G is generated by finite subgroups
each pair generates a group with property T and sufficiently large
constant then the whole group also has property T.
One can use this result to show that some groups like
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.
orbit closures and Diophantine approximations of
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
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.
|Expansion in SL(d, Z/qZ), q square-free.
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.