Second meeting, Technion, 13.3.14, Amado building room 619
On best possible rates of approximation
by lattice orbits on homogeneous varieties
We consider the orbits of lattice subgroups of semisimple groups acting on homogeneous spaces. We will give general lower and upper bounds on the rate of approximation of a general point by a generic orbit, and then give a sufficient criterion of when they match. The criterion is spectral in nature and holds quite often, leading to best possible results. This will be demonstrated in several examples.
Based on joint Work with A. Ghosh and A. Gorodnik
Zariski density in infinite dimension and applications to Furstenberg maps
Zariski density for a subgroup \Gamma of a semisimple Lie group G without compact factors can be rephrased geometrically using the symmetric space of G. We will see that such a result is also true for groups of bounded operators acting on infinite dimensional symmetric spaces of finite rank. In particular, there is a definition of infinite dimensional algebraic groups even if a Zariski topology does not really make sense.
We will see how it can give informations about Furstenberg maps for groups acting on symmetric spaces of finite rank leading to rigidity phenomena.
This a part of a joint work in progress with Uri Bader and Jean Lecureux.
Counting commensurability classes of hyperbolic manifolds
An hyperbolic manifold is a Riemannian manifold locally isometric to the n-dimensional hyperbolic space. Note that closed hyperbolic manifolds of finite volume correspond to torsion-free lattices in the group SO+(n,1).
An interesting direction in the study of such manifolds are counting questions:
By a classical result of Wang, in dimension >=4 there are finitely many isometry classes of hyperbolic manifolds up to any given finite volume V. More recently, Burger, Gelander, Lubotzky and Mozes showed that this number grows like V^V.
In this talk we focus on the number of commensurability classes of hyperbolic manifolds. Two manifolds belong to the same commensurability class if they admit a common finite cover. We show that in dimension at least 4, this number grows like V^V as well. This is a strengthening of the lower bound c^V established by Raimbault.
Since the number of arithmetic commensurability classes grows (almost) polynomially (Belolipetsky), our result implies that non-arithmetic manifolds account most" commensurability classes.
We will explain the ideas involved in the proof. The tools used are a mixture of arithmetic, hyperbolic geometry and some combinatorics.
This is a joint work with Tsachik Gelander.
Choices and Intervals
Consider the following point process on the unit circle. Finitely
many distinct points are placed on the circle in any arbitrary
configuration. This configuration of points subdivides the circle
into a finite number of intervals. At each time step, two points are
sampled uniformly from the circle. Each of these points lands within
some pair of intervals formed by the previous configuration. Add the
point that falls in the larger interval to the existing configuration
of points, discard the other, and then repeat this process.
We will study the behavior of a typical interval, and we will show
that as the number of points tends to infinity, this has an almost
sure limit, which we characterize.
This is joint work with Pascal Maillard (Weizmann).
The algebraic side of l2-Betti numbers
l2-Betti numbers are analytic objects and from their definition it remains mysterious why they are
rationally valued in most cases. In this survey talk we explore some of the reasons for that phenomenon.
We'll see some surprising connections to ring theory, algebraic K-theory and p-adic groups on the way.
You are cordially invited.