ACTION NOW WANDERING SEMINAR

Let G be a countable group, Ord(G) the (compact, metrizable) space of all the linear orders on G. The group naturally acts on this space from the left. A left invariant order on G is a fixed point for this action, and the group itself is called left orderable if such a fixed point exists. An invariant random order (IRO) is a Borel, G-invariant probability measure on Ord(G). This notion, defined by Alpeev, Meyerovitch and Ryu, is more flexible for example since every countable group admits an IRO.

In a joint work with Tom Meyerovitch and Yuqing Lin, we study this notion and show: While every countable group admits an IRO, we show that it is not true that any partially defined order can be extended to an IRO. We prove a Bauer-Poulsen dichotomy (a-la Glasner-Weiss) for the simplex of IRO's on a countable group. We obtain partial information on which ergodic actions can be realized by IRO's.

After introducing the notion of characters and their relation to classical rigidity results and Invariant Random Subgroups (IRS), we will show that the space of characters of the free group is a Poulsen simplex for every . We will also discuss the space of characters of other free products. This is based on joint work with Joav Orovitz and Itamar Vigdorovich.

A subset A of a discrete group G is called completely syndetic if for every positive integer n there are finitely many left translates of A such that every n elements of G belong together to at least one of these translates. In this talk, I will discuss the question in the title and present some relations to certain C*-algebras, Boolean algebras, and dynamical systems. In particular, I will explain how to construct nontrivial minimal proximal actions for non strongly amenable groups. I will also show how this machinery helps to characterize “dense orbit sets” answering a question of Glasner, Tsankov, Weiss, and Zucker.

The talk is based on joint work with Matthew Kennedy and Sven Raum and ongoing work with Ariel Yadin.

In this talk, I will present an effective equidistribution theorem (that is, with an explicit rate) for the action of a horospherical subgroup acting on the frame bundle of SO(n,1)/Gamma when Gamma is geometrically finite and has infinite covolume. We will discuss the difficulties that arise in studying these types of problems in this infinite volume, rank one setting, including the complicated behaviour of the (leafwise) Patterson-Sullivan (PS) measures. As key steps in the proof, we established quantitative nondivergence of horospherical orbits and built on the work of Das, Fishman, Simmons, and Urbanski to prove fundamental "friendliness" properties of the (leafwise) PS measures that are of interest more generally, and I will also talk about these results.

This is joint work with Nattalie Tamam.

A locally testable code is an error correcting code that has a property-tester who when receiving a word reads q bits of it that are randomly chosen, and rejects the word with probability proportional to its distance from the code. The parameter q is called the locality of the tester. In a joint work with Irit Dinur, Shai Evra, Ron Livne and Alex Lubotzky we construct an infinite family of locally testable codes which have constant rate, constant distance and constant locality.