# YOUNGCTION NOW!

YOUNGCTION NOW! No, it is not a traditional Chinese dish.
YOUNGCTION NOW! is the place where the young participants of the ACTION NOW! seminar gather, ask questions and wonder. Here, any question is welcome!
The meetings will be devoted to the wide area of group actions which includes ergodic theory and geometric group theory.

• 10:50 - 11:50 Rene Ruhr will talk about spectral gap, mixing and effective ergodic theorems.
We will discuss the notation of spectral gap and try to explain its usefulness by providing various effective ergodic theorems on homogeneous spaces.

• 12:00 - 12:50 Uri Bader will talk about the Howe-Moore theorem as a result about compactfication.
In my talk I will prove the famous Howe-Moore theorem regarding decaying of matrix coefficients of group representations. I will, in fact, explain how the theorem is equivalent to a result regarding compactifications, and prove the latter.

• Lunch break

• 14:00 - 14:50 Uri Bader

• Coffee break

• 15:20 - 16:50 Arielle Leitner will give an Introduction to Geometric Structures and Generalized Cusps on Convex Projective Manifolds.
The first half of the talk will be an introduction to geometric structures in the sense of Thurston. We will also review a bit of projective geometry. In the second part of the talk, we will discuss conditions for deforming properly convex projective structures to get new properly convex projective structures. A necessary condition is that the ends of the manifold have the structure of generalized cusps. I have classified these in dimension 3, and I have work in progress to classify in n dimensions with Sam Ballas and Daryl Cooper.

# Previously, on the YOUNGCTION NOW! meetings:

• 11:00 - 12:30 Oliver Sargent will give an Introduction to random walks on projective spaces.
We will consider random walks generated by linear actions on real projective spaces. We will show that understanding the asymptotic distribution of such walks is equivalent to understanding stationary measures. Under some assumptions there will be a unique stationary measure (this is a result due to Furstenberg) and we will elaborate on this case.

• 12:00-13:00 Lunch break

• 13:00-14:30 Uri Shapira will explain what is Decay of matrix coefficients and what is good about it.

• 14:30-14:45 Coffee again

• 15:00-16:30 Yaar Solomon will talk about Dense forests and the Danzer problem.
$Y\subset \mathbb{R}^d$ is a Danzer set if it intersects every convex set of volume 1. It is not known whether uniformly discrete Danzer sets exists or not (for $d>1$), and this straightforward question is open from the sixties. I will discuss a few weakening directions of that problem, focusing on dynamical argument that gives partial results.
Relevant references:
(-) Bambah, Woods - On a problem of Danzer ,
(-) Bishop - A set containing rectifiable arcs QC-locally but not QC-globally (pages 4-7),
(-) Solomon, Weiss - Dense forests and Danzer sets
• 10:00-11:00 Gil Goffer will talk about Uniform Structures on Simple Lie Groups.

• 11:15-12:45 Yair Hartman will explain what is the Furstenberg-Poisson boundary and its connection to amenability.
The talk will contain an introduction to the Furstenberg-Poisson boundary from the very beginning: motivations, definitions and applications.

• 13:15-14:45 Asaf Katz will describe some examples of effective equidistribution.
We intend to explain how one may deduce an effective equidistribution theorem by an effective formulation of a suitable ergodic theorem. We will focus on two cases - irrational rotation on the circle and the horocylic flow on the modular surface. In the former case - the effectivization is done by harmonic analysis, while in the later case, the effectivization follows from a mixing statement which relays on the Howe-Moore theorem and spectral gap, in particular, we will try to sketch proofs of Sarnak's theorem and Venkatesh's sparse equidistribution theorem.
• 10:30-12:00 Construct your very own CAT(0) cube complex - the DIY guide by Nir Lazarovich.
We will talk on connection between recurrence property in dynamical systems and combinatorial structures inherent in the densely large set of the set of natural numbers. Especially we will present Sarkozy's theorem, which says that if a set is large, then the set of difference is combinatorially rich. The original proof was obtained by analytic number theory methods, but here we will discuss a different ergodic approach to this result to enable us to obtain new extensions of Sarkozy's theorem.

• 13:00-14:30 Anton Malyshev will construct groups of intermediate growth.
Groups of intermediate growth are finitely generated groups in which the number of elements of word length n grows faster than any polynomial function of n, but slower than any exponential function of n. We will construct the most famous example, the Grigorchuk group. This construction and similar ones are a good source of counterexamples in group theory. The Grigorchuk group is the first example on an amenable group which is not elementary amenable. It is also an infinite torsion group.

• 14:45-15:45 Gidi Amir will talk about Automata groups, permutational wreath products and inverted orbits.
I will discuss some constructions of automata groups and permutational wreath products over automata groups and briefly outline how these are used to construct groups where one can carefully control some parameters such as speed of random walks and word growth. We will see how these properties are related to the "inverted orbit" of the action of the group.
• 10:00 - Youngwon Son will talk about Sets of recurrence.
We will talk on connection between recurrence property in dynamical systems and combinatorial structures inherent in the densely large set of the set of natural numbers. Especially we will present Sarkozy's theorem, which says that if a set is large, then the set of difference is combinatorially rich. The original proof was obtained by analytic number theory methods, but here we will discuss a different ergodic approach to this result to enable us to obtain new extensions of Sarkozy's theorem.

• 11:30 - Tsachik Gelander will count hypergolic manifolds.

• 12:45 - Arie Levit will count commensurability classes of hyperbolic manifolds.
Two hyperbolic manifolds are commensurable if they admit a common finite cover. We will discuss the question of counting the number of such manifolds up to a given volume and up to commensurability. We will explain the relation of this to other interesting counting questions. The proof relies on a mixture of geometry, combinatorics and number theory.
In the spirit of the YOUNGCTION NOW! we will try to keep the talk as self-contained as possible.
This is a joint work with Tsachik Gelander.
• 10:30 - Kiran Parkhe will talk about actions of lattices on manifolds.
An action of a group $G$ on a manifold $M$ is a continuous homomorphism $\phi \colon G \to Homeo(M)$, the group of homeomorphisms of $M$. The "Zimmer program" asks (in analogy with work of Margulis on linear representations): If $L$ is a lattice in the Lie group $G$, are actions of $L$ on manifolds "rigid" in the sense that they are the restrictions of actions of $G$?
This program has proven to be quite difficult. For example, actions of $SL(3, \mathbb{R})$ on surfaces are well understood (and quite limited). We conjecture that the only actions of $SL(3, \mathbb{Z})$ on surfaces are restrictions of these, but this is not known.
In this talk, we prove one of the few things that is known for surface actions of $SL(3, \mathbb{Z})$: any action by $C^1$ diffeomorphisms, having a point fixed by every element, is trivial. This may sound intimidating, but the pieces are actually easy and illuminating.

• 13:00 - Miel Sharf will talk about Algebraic representations of ergodic actions.
Consider an ergodic action $S \curvearrowright X$. Can one somehow identify $S$ and $X$ as algebrao-geometric objects? We define what is an algebraic representation of such action $S \curvearrowright X$ and show that these form a category.
We also show that this category has an initial element and analyse a case in which this initial object is trivial, meaning that any way to represent the action $S \curvearrowright X$ in an algebraic way is abysmal - almost all points of $X$ are identified under this representation.

I will recall and discuss the celebrated Margulis super-rigidity.
I will prove it using the tools developed in Miel's talk.
At Tel-Aviv University, Schreiber bld. (math department), Room 8.

• 10:00 - A lecture by Doron Puder about Words Maps and Measure Preservation.
Given a word $w$ in the free group $\mathbb{F}_k$ and a group $G$, consider the map $w:G\times G\times \dots\times G \to G$ from the cross product of $k$ copies of $G$ to $G$, defined by substitutions. In recent years there has been much interest in word maps on groups, with various motivations and applications. We shall survey several problems and results in this theory, and focus on questions regarding the image of this map on certain finite groups as well as the distribution, or measure, induced by a given $w$ on these groups.

• 12:15 - Erez Nesharim - Badly approximable numbers and absolute winning sets.
The set of badly approximable numbers is a meagre set that have zero Lebesgue measure, but yet has the surprising property that the intersection of every countably many translations of it is uncountable. W.M.Schmidt discovered in 1965 a powerfull tool now called Schmidt's game, which he used to reprove the above. We present a variation of this game, now called the absolute game, and give two proofs that the set of badly approximable numbers is a winning set for this game.Badly approximable numbers and absolute winning sets.

• 14:15 - Steffen Weil - Bounded geodesics on hyperbolic manifolds.
Continuing with Schmidt games, we study bounded geodesics (in various contexts) on a (finite volume) complete hyperbolic manifold. I will draw a lot of nice pictures and try to keep the talk at a basic level. As far as time permits necessary background and proofs will be given. Moreover, I might discuss further connections/applications to Diophantine approximation.

At Ben Gurion University, room -101, in Deichmann Building (#58)

• 10:30 - Dennis Gulko will talk about Sharply 2-transitive and 3-transitive linear groups.
Basic definitions and examples of sharply 2-transitive groups will be introduced. We will introduce the affine action and will show a sketch of proof that under some assumptions this is the only example of sharply 2-transitivity. Later we will talk about sharply 3-transitive groups and present some interesting results.

• 13:00 - Idan Perl will talk about Gromov's Theorem part I: Kleiner's theorem.
Let $G$ be a group of weakly polynomial growth, and positive integer $d$. Then the space of harmonic functions on $G$ with polynomial growth of at most $d$ is finite dimensional.

• 15:00 - Ariel Yadin will talk about Gromov's Theorem part II: From Kleiner to Gromov.
We will try to get from Kleiner's Theorem (every group of polynomial growth has a finite dimensional space of Lipschitz harmonic functions) to Gromov's Theorem (every group of polynomial growth is virtually nilpotent).
I will try to explain the basic induction argument. The important part is to understand why one wants to find a homomorphism $G \to \mathbb{R}$, and why a finite dimensional space of Lipschitz harmonic functions gives such a homomorphism.
At The Hebrew University (room B221, middle building in the new Computer Science department)

• 11:00 - Max Gurevich will talk about Spherical functions on p-adic groups.
Zonal spherical functions on a classical symmetric space such as a sphere or a plane are long known to have a special role in the spectral analysis of the space. More generally, spherical functions come attached to a group (classically, a connected Lie group). This collection of functions can serve as a parametrization for the unramified inifinite-dimensional representations of the group, or as the spectrum of the so-called spherical Hecke algebra. A non-trivial isomorphism (work of Harish-Chandra in the Lie group setting, and of Satake for p-adic groups) identifies that spectrum with a simple complex domain. All these notions have a long history of significance in representation theory, including the central role of the above Satake isomorphism in the far-reaching functoriality conjectures of the Langlands program.
In this talk I will attempt to give an introductory survey on the subject and on the p-adic setting in general.

• 13:30 - Ayala Byron will talk about Structure theory for groups acting on graphs: Bass-Serre theory.
We will review free products with amalgamations and HNN extensions of groups, and generalize to graphs of groups and the fundamental group of such. Then we'll see how such groups act on trees, and why an action of a group on a tree (with no fixed points) is the same as a splitting of the group as the fundamental group of a graph of groups.

• 15:15 - Zlil Sela will talk about canonical decompositions of groups and some of their applications.
Grushko's theorem from 1939 associates a canonical free decomposition with any given finitely generated group. We intend to describe the canonical JSJ decomposition of a finitely presented group, that describes more general decompositions. We will explain the connection of this decomposition with low dimensional topology, and present some of its basic applications.
At Weizmann (room 261, Ziskind building)

• 13:30 - Itai Benjamini will talk about Invariant random stuff.
We will discuss invariant metrics, partitions and percolation on Cayley graphs.

• 14:45 - Open discussion about amenability and related open problems

• 16:00 - Elliot Paquette will talk about Kazhdan's property for (T)oddlers.
Kazhdan's property (T) is a group rigidity property that comes from the theory of representation theory. Since its definition, it has seen use in many areas of math, in particular in ergodic theory and geometric group theory. Informally, it can be considered as a type of strong non-amenability. We will begin by establishing the basic theory of property (T) and its connection to amenability. We will then show its connections to graph expansion and some recent work on random groups.
At Tel Aviv university (Room 110, Ornstein building), Joint with the student seminar in ergodic theory in TAU.

• 14:00 - Yair Hartman will give an Introductory talk on Amenability
There are plenty of equivalent definitions to the notion of amenable group. These definitions are stated in the language of geometry of groups, group actions, representation theory, analysis and more. This variety hints that this concept is important from many points of view. We will discuss these equivalent definitions of amenability, prove basic properties and talk about some extensions of this concept.
In the next YOUNGCTION NOW! meeting we will have a talk on the related notion of property (T), which is in some sense on the opposite extreme of amenability.

• 15:30 - Coffee and cookies!

• 16:00 - Rodrigo Trevino will give A soft introduction to flat surfaces, Veech groups, and ergodic properties of translation flows.
Most people know what the 2-torus is, that it has a flat metric, and that you can define on it a linear (translation) flow. Moreover, most people know that the properties of this flow satisfies a very nice dichotomy: the flow is either periodic (if the slope is rational) or unique ergodic (if not). Luckily, there are other flat surfaces you can consider that are not the torus. These are higher genus surfaces (infinite genus is allowed) which have a flat metric almost everywhere and where you can define a linear (translation) flow just the way you do it for the torus. You can ask yourself "When is a flat surface like the torus?" and there are many ways of trying to answer this, but a very nice way to approach it if your surface has a non-trivial Veech group, which is some sort of group of symmetries.
I'll introduce things, present lots of cute examples, and convince you that this is interesting.
• 9:30 - Coffee and cookies in the math department

• 10:00 - Arie Levit will give an Introductory talk on Invariant Random Subgroups (IRSs), in room A, Feinberg building
An invariant random subgroup is a probability measure on the space of closed subgroups of a given group, which is invariant under conjugation. Such an object can be thought of as a generalization of both normal subgroups and lattice subgroups, and is a natural object to study when one deals with group actions on probability spaces. Our goal is to discuss the basic definitions, e.g. the Chaubuty topology on the space of closed subgroups, examples and constructions of IRSs. If time permits we will discuss some more advanced results, e.g. the fact that every IRS can be associated with a group action and a generalization of the Borel density theorem to IRSs (these appear in this paper).