Activities This Week

In this talk I will provide some counter-examples for quantitative multiple recurrence problems for systems with more than one transformation. For instance, I will show that there exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $\ell < 4$ there exists $A\in \mathcal{X}$ such that $\mu(A\cap T_1^n A\cap T_2^n A) < \mu(A)^{\ell}$ for every $n \in \mathbb{N}$. The construction of such a system is based on the study of ``big’’ subsets of $\mathbb{N}^2$ and $\mathbb{N}^3$ satisfying combinatorial properties.

This a joint work with Wenbo Sun.

Let G be a semisimple Lie group of rank two or higher. We discuss certain asymptotic properties for sequences of lattices inside G.

A lattice in G is associated to a classical geometric object of the form M = K\G/Gamma. We allow G to be either real or p-adic. An important geometric property for such sequences of metric spaces is Benjamini-Schramm (BS) convergence. We present a theorem saying that any sequence of distinct M’s is BS-convergent.

It turns out that the geometric notion of BS-convergence has implications to representation theory, in terms of Plancherel measure convergence, and to topology, in terms of convergence of normalized Betti numbers. We will briefly mention these implications.

Kazhdan’s property (T) plays an important role in the above results. We will explain a novel approach relying on Selberg’s property instead and extending to products of rank one groups (such as SL2xSL2).

The talk is based on [Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet] and two recent preprints by Gelander-L. and L.