תורת החבורות וגיאומטריה Ical Atom

I will prove that an RF group has property (T) if and only if any of it‘s associated box spaces have geometric property (T).

This will be an organizational meeting for this new seminar. Please bring or send your time constraints.

Time premitting I will also give a short survey talk: Uniformly recurrent subgroups and groups with a trivial amenable radical that fail to be $C^*$-simple. In this I will describe the new construction by Adrien Le-boudec of such groups, following the criterion for $C^*$-simplicity by Kalantar-Kennedy.

One way of viewing coarse geometry is that it is the perfect tool for treating metric spaces like groups and doing representation theoretic/harmonic analytical techniques in a much larger setting. R. Willett and G. Yu define a property (T) for coarse spaces using the uniform Roe algebra. In this talk, I plan to define coarse spaces and show that the uniform Roe algebra is nice tool that acts like the group C* algebra. I will define geometric property (T) and discuss some of its properties.

In this talk I‘ll present a joint work with Ariel Yadin, in which we solve the Furstenberg Entropy Realization Problem for finitely supported random walks (finite range jumps) on free groups and lamplighter groups. This generalizes a previous result of Bowen. The proof consists of several reductions which have geometric and probabilistic flavors of independent interests.

All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.

We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin and Navas.

A discrete set $Y$ in $R^d$ is called a dense forest if for every positive $\epsilon$, $Y$ is epsilon close to all line segments of length $V(\epsilon)$, for some function $V(\epsilon)$. We will discuss the intuition of this definition and the motivation for having such sets. Then I will present three constructions for dense forests by Bishop-Peres, S.-Weiss, and by Alon, that use basic Diophantine approximations, homogeneous dynamics, and the Lovasz local lemma, respectively. The focus will be on our result (jointly with Barak Weiss) for which I hope to give all the details of the construction. All the definitions and the background will be given in the talk.

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. The most intuitive example is to imagine blowing up a sphere so that eventually it becomes so large, it looks like a plane. This is a transition from spherical geometry to Euclidean geometry.

We will study limits of the Cartan subgroup in $SL(n,R)$. A limit group is the limit under a sequence of conjugations of the Cartan subgroup in $SL(n,R)$. We will show using the hyperreal numbers that in $SL(3,R)$ there are 5 limit groups, each determined by a degenerate triangle.

In the second part of the talk, we will show that for $n \geq 7$, there are infinitely many nonconjugate limit groups of the Cartan subgroup.

Actions of countable discrete groups $\Gamma$ on a compact (metrizable) group $X$ by (continuous) group automorphisms are a rich class of dynamical systems. The case where $X$ is abelian is an important subclass, also called ”algeraic actions“. By Pontryagin duality, algebraic actions are in one-to-one correspondence with $\mathbb{Z}\Gamma$-modules. There is a fascinating ”dictionary“ between the two, a beautiful interplay between dynamics, Fourier analysis, and commutative or noncommutative algebra. In the last several years, much progress has been made towards understanding the algebraic actions of general countable groups. Somehow surprisingly, operator algebras turn out to be important for such a study. This introductory talk will cover some basic aspects of the theory. (New results and open questions might be discussed in a followup talk).

Sharply 2-transitive groups are groups that admit a transitive and free action on pairs of distinct points. Finite sharply 2-transitive groups have been thoroughly studied, and completely classified by H. Zassenhaus in the 1930‘s, but up to some few years back, relatively little was known about the infinite case. In this lecture we will survey the latest developments regarding infinite sharply 2-transitive groups, and present our results in this field.