ACTION NOW WANDERING SEMINAR

Second meeting, Weizmann Institute, 2.1.19

Which groups have bounded harmonic functions?

Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all abelian groups, and more generally, virtually nilpotent groups are "Choquet-Deny groups": these groups cannot support non-trivial bounded harmonic functions. Equivalently, their Furstenberg-Poisson boundary is trivial, for any random walk. I will present a recent result where we complete the classification of discrete countable Choquet-Deny groups, proving a conjecture of Kaimanovich-Vershik. We show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key here is not the growth rate, but rather the algebraic infinite conjugacy class property (ICC). This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.

Temporal distributional limit theorems for irrational rotations (joint with D. Dolgopyat)

A temporal distributional limit theorem is a scaling limit for the distributions of the ergodic sums at random time t between 1 and N, and fixed initial condition x, as N tends to infinity. Bromberg and Ulcigrai showed that such limit theorems hold for "nice" observables for a.e. x for all irrational rotations of bounded type (a set of angles with full Hausdorff dimension). The result is that they do not hold for a set of angles with full Lebesgue measure.

Afternoon session at the particle accelerator auditorium

Surface subgroups in uniform lattices of some semisimple Lie groups

In a joint work with Jeremy Kahn and Francois Labourie we prove that any uniform lattice in a simple complex Lie group G contains a surface group. (I.e. the fundamental group of an orientable surface of genus at least 2).

Flexible Stability of Surface Groups

Roughly speaking, a finitely presented group is said to be (flexibly) stable if any approximate action of the group on a finite set is an approximation of an action. Stability is closely related to local testability (in CS), soficity, residual finiteness, and property (T). In this joint work with Arie Levit and Yair Minsky, we show that surface groups are flexibly stable using the geometry of CAT(0) spaces and a new quantitative variant of LERF.

A crash course on Ratner's theorems

Ratner's theorems, proved in the early 1990s, were a breakthrough in dynamics, making it possible to apply ergodic theory to many problems in number theory, geometry, etc. I will discuss the statements of the theorems and highlight two ingredients of the proofs: the R property and use of generic points.

Japanese pianist Haruka Yabuno, the double bass player Ehud Ettun and drummer Tsachik Gelander.

For more on the music of Haruka Yabuno and Ehud Ettun, click here and here .