ACTION NOW WANDERING SEMINAR

Second meeting, Hebrew University, 30.1.18, Feldman Building (Institute for Advanced Studies) room 130

Dynamics, arithmetic progressions, and approximate cohomology

Stability and Invariant Random subgroups

Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation XY=YX, and turns out to be equivalent to the following property of the group Z^2=: Every "almost action" of Z^2 on a finite set is close to a genuine action of Z^2. This leads to the notion of stable groups.

We will describe a relationship between stability, invariant random subgroups and sofic groups, giving, in particular, a characterization of stability among amenable groups. We will then show how to apply the above in concrete cases to prove and refute stability of some classes of groups. Finally, we will discuss stability of groups with Kazhdan's property (T), and some results on the quantitative aspect of stability.

Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.

Discretized sum-product estimates in higher dimension

Bourgain's discretized sum-product theorem asserts that if a subset of real numbers satisfies some non-concentration property, then its size grows significantly when it adds and multiplies to itself element-wise. In this talk I will discuss higher-dimensional analogues of this phenomenon, namely sum-product estimates in simple algebras and in Euclidean spaces under the action of linear endomorphisms.

On rational planes in four-dimensional space and their associated lattices

To any two-dimensional rational plane in four-dimensional space, one can naturally attach a point in the Grassmannian $\operatorname{Gr}(2,4)$ and four lattices of rank two (each corresponding to a point on the modular surface). Here, the first two originate from the lattice of integer points in the plane and its orthogonal complement and the second two essentially arise from the accidental local isomorphism between $\operatorname{SO}(4)$ and $\operatorname{SU}(2)^2$.

We prove the simultaneous equidistribution of all of these objects, which is an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings on the product of rank one groups. I will explain the natural construction of these associated four lattices of rank 2, formulate an equivalent equidistribution statement in homogeneous dynamics and explain how the latter follows from Einsiedler-Lindenstrauss result.

This is a joint work with Manfred Einsiedler and Andreas Wieser.