ACTION NOW WANDERING SEMINAR

First meeting, Ben Gurion University, 26.11.17, Building 58 room -101

Equidistribution of divergent orbits in the space of lattices

A well known application of the pointwise ergodic theorem (PET) states that almost every x in [0,1] has a normal continued fraction expansion, or equivalently its orbit under the Gauss map T(x):={1/x} (where {y} is the fractional part of y) equiditributes with respect to the Gauss Kuzmin measure dt/( ln(2)(1+t) ). This is not true for all x, and in particular it fails for rational numbers which have finite continued fraction expansions.

In this talk we shall see how to "extend" the PET to rational numbers and its connection to divergent orbits of the diagonal group in the space of 2-dimensional lattices. Furthermore, we shall show how the natural setting of this problem is actually over the adeles, and in particular it can be formulated in any dimension (for which we give some partial results).

This is a joint work with Uri Shapira from the Technion.

Stable and well-rounded lattices in diagonal orbits

For the action of the diagonal group A on the space of lattices SL(n,R)/SL(n,Z), it is shown that every A-orbit contains both a well-rounded lattice and a stable lattice.

Entropy, metrics and quasi-morphisms

One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots.

Let S be a compact oriented surface. In this talk I will discuss several invariant metrics and quasi-morphisms on the identity component Diff_0(S, area) of the group of area preserving diffeomorphisms of S. In particular, I will show that some quasi-morphisms on Diff_0(S, area) are related to the topological entropy. More precisely, I will discuss a construction of infinitely many linearly independent quasi-morphisms on Diff_0(S, area) whose absolute values bound from below the topological entropy. If time permits, I will define a bi-invariant metric on this group, called the entropy metric, and show that it is unbounded. Based on a joint work with M. Marcinkowski.

Stability of asymptotic representations, cohomology vanishing, and non-approximable groups

First of all, I plan to survey various approximation properties of discrete groups and mention a few applications to the theory of groups and group rings. We provide the first examples of countable groups which are not Frobenius-approximated. Our strategy is to use higher-dimensional cohomology vanishing phenomena to prove that any Frobenius-almost homomorphism into finite-dimensional unitary groups is close to an actual homomorphism and combine this with existence results of certain non-residually finite central extensions of lattices of higher rank. We ultimately rely on work of Garland, Ballmann-Świątkowski, Deligne, Rapinchuk and others.

Barak Weiss, Tel Aviv University

The illumination problem. The following elementary problem in geometry is still open: given a polygon P in the plane, say that points x and y in P see each other if there is a billiard path from x to y. Is there a polygon in which infinitely many pairs of points do not see each other?

Such problems turn out to be easy to state but very difficult to solve. I will explain this and related questions in greater detail, and describe some recent progress which relies on well-known work of Eskin and Mirzakhani (part of the late Maryam Mirzakhani's Field medal citation).

Jazz show

Ehud Ettun - base

Haruka Yabuno - piano

Tsachik Gelander - drums