BGU Probability and Ergodic Theory (PET) seminar Ical Atom

Cornulier conjectured that two completely solvable Lie groups are quasiisometric if and only if they are isomorphic. This is a very difficult and very open problem. In this talk I will present some of the structure theory of solvable Lie groups and focus on the importance of sublinear distortions to this theory. I will review some of the important work of Cornulier and Tessera on Dehn functions of these groups. Finally, I will present recent results that contribute to their QI classification, which are interesting (also) because they are based on a (sublinear) weak form of QI.

Based on joint work with Gabriel Pallier.

The notion of a proper Ellis semigroup compactification is introduced. Using Ellis’s functional approach their connection with equiuniformities on a topological group is established. Proper Ellis semigroup compactification of a topological group G from the maximal equiuniformity on a phase space in the case of isometric action (on a discrete space, on a discrete chain, as liner isometries of a Hilbert space) is described. Its connection with Roelcke uniformity on G is established.

Kac’s lemma is a classical result in ergodic theory. It asserts that the expected number of

iterates that it takes a point from a measurable set A to return to the set A under an ergodic

probability-preserving transformation is equal to the inverse of the measure of A. As we will discuss in this seminar, there is a natural generalization of Kac’s lemma that applies to probability preserving actions of an arbitrary countable group (and beyond). As an application, we will show that that any ergodic action of a countable group admits a countable generator. The content of this work is based on a joint article with Benjamin Weiss

https://doi.org/10.48550/arXiv.2410.18488

A Lipschitz function on a graph G is a function f:V->Z from the vertex set of the graph to the integers which changes by at most 1 along any edge of the graph. Given a finite connected graph G, and fixing the value of the function to be 0 on at least one vertex, we may sample such a Lipschitz function uniformly at random. What can we say about the typical height at a vertex? This depends heavily on G. For example, when G is a path of length n, and the height at one of the endpoints is fixed to be 0, this model corresponds to a simple random walk with uniform increments in {-1,0,1}, and hence the height at the opposite endpoint of the path is typically of order sqrt(n). In this talk, we consider the case when G is a d-regular tree of depth n, and the height at the leaves is fixed to 0. Peled, Samotij and Yehudayoff showed that the height at the root of the tree is tight as n grows, having doubly exponentially decaying tails. We study the question of whether the distribution of the height at the root converges as n tends to infinity. It turns out that the answer depends on d, with a phase transition occurring between d=7 and d=8. We explain the reasons for this and outline some details of the proof. Joint with Nathaniel Butler, Kesav Krishnan and Gourab Ray.

We introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given topological dynamical systems $(X, T), (Y, S)$, and a continuous function $f \in C(X \times Y)$, what can be said about the extrema $\sup_{y \in Y} \inf_{x \in X} \lim_{k \to \infty} \frac{1}{k} \sum_{j = 0}^{k - 1} f \left( T^j x , S^j y \right)?$

I will demonstrate how careful analysis of group relations yields unexpected constructions, addressing several central questions in group theory. These include a question by Elliott, Jonusas, Mesyan, Mitchell, Morayne, and Peresse regarding Zariski topologies on groups and semigroups, a series of questions by Amir, Blachar, Gerasimova, and Kozma concerning algebraic group laws, and a longstanding question by Wiegold on invariably generated groups.