The seminar will take place in-person on Wednesdays 2PM (CET) .  The topics covered include: locally symmetric spaces,  homogeneous dynamics, representation theory, geometric and measured group theory. 

Past meetings

• Lasse Wolf (IHES)

Oct 29th 12:00 (new time!)  Faculty of Mathematics and Computer Science, Conference Hall B (under the library)

Title: Discrete subgroups of slow growth and their connection to temperedness

Abstract: For a discrete subgroup $\Gamma$ of a semisimple Lie group $G$ with finite center, one is interested in properties of the regular representation $L^2(\Gamma\backslash G)$, the spectral theory of the locally symmetric space $\Gamma\backslash G /K$, and the growth rates of $\Gamma$ in $G$. In this talk, I will describe these three aspects and the connections between them. In particular, I will characterize the temperedness of the regular representation in terms of Quint’s growth indicator function. Temperedness has been conjectured to hold if $\Gamma$ is a Borel Anosov subgroup, and I will sketch how one can prove this conjecture in most cases.

• Minju Lee (University of Chicago)

July 9th, 14:00  (Faculty of Mathematics and Computer Science, Conference Hall B (under the library)

Title : Horocycles in geometrically finite rigid acylindrical manifolds 

Abstract: The Ratner-Shah theorem states that the closure of a horocycle in a finite-volume hyperbolic 3-manifold is always a properly immersed submanifold. Classifying horocyclic orbit closures beyond the classical setting remains an important problem. Following the pioneering work of McMullen-Mohammadi-Oh on generalizing the Ratner-Shah theorem to infinite-volume manifolds, there have been many exciting developments. In this talk, I will present joint work with Dongryul Kim, in which we classify horocyclic orbit closures for a class of geometrically finite 3-manifolds and discuss how our results fit into the broader literature.

• Spyridon Petrakos (University of Munster) 

May  28th, 14:00 (Faculty of Mathematics and Computer Science, room 1009).

Title: The small boundary property and classifiability of crossed products

Abstract: The small boundary property—a dynamical/topological property introduced in the late '90s by Lindenstrauss and Weiss in relation to mean dimension, topological entropy, and shift embeddability—has somewhat recently taken a prominent role in the classification theory for crossed product C*-algebras. In this talk, we will give a gentle primer to the latter, introduce the SBP, and discuss criteria for it to hold. Based on joint work with David Kerr and Grigoris Kopsacheilis.

• Colin Jahel (TU Dresden)

May  21st, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title:  Measured dynamics of non-Archimedian Polish group

• Abstract:  The study of group dynamics, i.e. the study of their continuous or measured actions, is classically focused on locally compact groups, such as free groups or special groups SL_n(R). My goal in this talk is to present a class of non-locally compact groups with interesting dynamical properties: non-Archimedian groups, i.e. closed subgroups of the permutation group of N. I will explore topics of amenability, rigidity of pmp actions and invariant random subgroups, in order to display the interesting behaviors we encounter in the realm of non-Archimedian groups.
  
  

• László Márton Tóth (Renyi Institute)

May  14th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Towards a limit theory of matroids

Abstract: Matroids are discrete structures that abstractly generalize the notion of independence in linear vector

spaces. They were introduced in 1935 by Whitney, and their research is very active to this day. They

provide a framework for understanding and solving various combinatorial and optimization

problems, and their study has led to efficient algorithms in network design and

operations research. They can be defined equivalently through various sets of axioms, describing their

independent sets, their bases, or, most notably for our purposes, through their submodular rank function.

In 2023 Lovász started the development of a limit theory by analyzing submodular set functions in an analytic setting. He also introduced matroids associated to graphings. I aim to report on the progress made since then by introducing a convergence notion, exploring connections to graph limits, and extending finite combinatorial results on fractional spanning trees and Whitney's "rigidity" theorem to a measurable setting. I will not assume any familiarity with matroids.

• Mark Pengitore (IMPAN)

May  7th, 14:00 (Faculty of Mathematics and Computer Science, room 1009).

Title: Growth Functions and linearity of automorphism groups of hyperbolic groups

Abstract: This talk will introduce various growth functions associated to a finitely generated group which measure the difficulty of separating an element from the identity using epimorphisms to a fixed family of nonabelian finite simple groups with characteristic kernels as a function of the word length. As an application of these functions, we provide a characterization of when the automorphism group of a hyperbolic group is linear.

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• Eduardo Silva (University of Munster)

Mar  12th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Continuity of asymptotic entropy on groups

Abstract: The asymptotic entropy of a random walk on a countable group is a non-negative number that determines the existence of non-constant bounded harmonic functions on the group. A natural question to ask is whether the asymptotic entropy, seen as a function of the step distribution of the random walk, is continuous. In this talk, I will explain two recent results on the continuity of asymptotic entropy: one for groups whose Poisson boundaries can be identified with a compact metric space carrying a unique stationary measure, and another for wreath products A ≀ Z^d, where A is a countable group and d ≥ 3.

• Matt Bowen (Oxford)

Mar  5th, 14:00 (Faculty of Mathematics and Computer Science,  Conference Hall B). 

Title: Measurable matchings in amenable group actions

Abstract:  Let $\Gamma$ be a finitely generated amenable group which is not virtually $(\mathbb{Z},+), and $G$ the Schreier graph of a free Borel action of $\Gamma$ on a standard probability space.  We show that if $G$ is bipartite, then $G$ has a measurable perfect matching.  This generalizes and gives a shorter proof of a result of the speaker, Kun, and Sabok, who proved the same result in the special case of pmp actions.  This talk is partially based on joint work with Poulin and Zomback and Kun and Sabok.

• Piotr Mizerka (Adam Mickiewicz University, Poznań)

Feb 19, 14:00  (Faculty of Mathematics and Computer Science, room 1009).  

Title: "Non-vanishing of group cohomology of SL(n,ℤ) in the rank"

Abstract: It has been recently shown by Bader and Sauer that H^k(SL(n,ℤ);π) = 0 for k <= n − 2 and π being a unitary representation without non-zero invariant vectors. We show for n = 3 and n = 4 the existence of orthogonal representations π of SL(n,ℤ) which do not have nontrivial invariant vectors and for which H^{n−1}(SL(n,ℤ);π) is nonzero. This proves that the Bader-Sauer result is sharp for these degrees. This is the joint work with B. Brück, S. Hughes and D. Kielak.

• Antonio López Neumann (IMPAN)

Feb 5th, 13:00 (Faculty of Mathematics and Computer Science, room 1009).  Note the modified time, the talk will be followed by short student talks on various topics around geometric group theory and dynamics. 

Title: On L^p-cohomology of semisimple groups

Abstract: L^p-cohomology is a rather fine quasi-isometry invariant popularized by Gromov. In the case of semisimple groups, he conjectured that it should behave in a classical way in degrees up to the rank (ie vanishing below the rank for all p>1 and non-vanishing at least for some p in the rank). This talk will introduce L^p-cohomology and present known results pointing towards this conjecture. We will then focus on two results, namely vanishing in degree 2 for most semisimple groups of rank at least 3, and non-vanishing in degree equal to the rank for affine buildings.

• Subhadip Dey (MPI Leipzig)

Jan  29th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Discrete subgroups of Lie groups with full limit sets

Abstract: In this talk, we'll discuss the extent to which a discrete subgroup of a higher-rank Lie group G with a full limit set in its Furstenberg boundary must be a lattice. We will demonstrate the existence of non-lattice discrete subgroups of G with full limit sets; these groups are infinitely-generates and constructed via an appropriate sequence of Anosov subgroups. This result raises the intriguing question of whether finitely generated examples exist. If time allows, we will discuss some work in progress related to this. This talk will based on joint work with Sebastian Hurtado.

• (expository talk) Hector Jardón Sanchez (UJ) 

Dec 11th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title SL(3,Z) has Property (T)

Abstract: The aim of this talk is to prove that the group SL(3,Z) has Property (T), following an argument of Yehuda Shalom. This talk is a complement to the last lecture of Mikołaj's course. Beyond the trivial example of compact groups, proving that a group has Property (T) is not, in general, a simple task.

We will begin the talk introducing Relative Property (T). Relative Property (T) is a rigidity property for representations of a group with respect to a subgroup. We will prove relative Property (T) for the pair $(SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2,\mathbb{Z}^2)$. Relative Property (T) of the latter pair, combined with the fact that SL(3,Z) is boundedly generated by elementary matrices concludes the proof.

At the end of this meeting we will list possible topics for short talks by students who need a grade for this seminar.

• AF/NF seminar (Dec 2nd), see https://sites.google.com/view/afnt-seminar/home

• Márton Borbényi (Renyi Institute, Budapest)

Nov 20th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Random interlacement is a factor of i.i.d.

Abstract:  The random interlacement is a Poisson point process on the space of labeled doubly infinite trajectories on a transient graph $G$. We show that the random interlacement is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by the vertices of the graph in some automorphism invariant way.

Our proof uses a variant of the soft local time method to construct the interlacement point process as a point-wise limit of a sequence of finite-length variants of the model with increasing length.  We also discuss a more direct method that works if and only if $G$ is non-unimodular.

• Ben Lowe (University of Chicago)

Nov 13th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Minimal submanifolds, higher expanders, and waists of locally symmetric spaces

Abstract: Gromov initiated a program to prove statements of the following form: Suppose we are given two simplicial complexes X and Y, where X is "complicated" and Y is "simple". Then any map f: X-> Y must have at least one "complicated" fiber.  In this talk I will describe various results of this kind for finite volume locally symmetric spaces, that are proved by bringing new tools into the picture from minimal surface theory and representation theory.  Based on joint work with Mikolaj Fraczyk.  

• Antoine Poulin  (McGill University, Montreal)

Oct 30th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Measure equivalence of Baumslag Solitar groups and type III relations

Abstract:  In this talk, we establish the measure equivalence classification of the Baumslag Solitar groups. We study the last remaining case, whose quasi-isometry analogue was solved in 2001. The crucial technique is that of type III equivalence relations, those who do not preserve measure. We will define them and present a few properties and invariants.

• Hector Jardón Sanchez  (Jagiellonian University, Kraków)

Oct 23th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: A Niblo-Roller Theorem for measured Property (T)

Abstract:  The Niblo-Roller Theorem is an expression of geometric rigidity for groups with Kazhdan's Property (T). This theorem states that any action of a Property (T) group on a CAT(0) cube complex has a fixed point. In the talk, we will discuss a generalization of the Niblo-Roller Theorem for pmp cbers with measured Property (T). This theorem implies that wallings on pmp cbers with measured Property (T) satisfy certain rigidity properties. If time allows, we will discuss applications of this result. The content of the talk is part of the speaker's thesis, recently submitted.

• Greg Terlov  (Renyi Institute, Budapest)

Oct 16th, 14:00 (Faculty of Mathematics and Computer Science, room 1009). 

Title: Quasi-pmp Borel graphs, weighted-amenability, and percolation

Abstract: Measured group theory and percolation theory on transitive graphs are closely related, but both fields are mainly developed under additional assumptions: measured group theory focuses on the pmp setting, while percolation is mostly studied on unimodular graphs. These assumptions are in many ways parallel, and extending beyond them requires similar techniques. Recently, there has been a lot of progress in understanding the measure class preserving setting. This talk will explore developments on the percolation side of the interplay. We do so by considering a notion of weighted-amenability for graphs that coincides with the amenability of its automorphism group, introduced by Benjamini, Lyons, Peres, and Schramm in 1999. We will discuss its relation with hyperfiniteness, establish an extension of Kesten’s spectral radius theorem, characterize via invariant random spanning trees and forests, and derive results about percolation phase transitions. This is a joint work with Ádám Timár.

• Krzysztof Święcicki (Wrocław University of Science and Technology)

May 9, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title:  No dimension reduction for doubling spaces of $\ell_q$ for $q>2$

Abstract: I'll discuss a new elementary proof for the impossibility of dimension reduction for doubling subsets of $\ell_q$ for $q>2$. This is done by constructing a family of diamond graph-like objects based on the construction by Bartal, Gottlieb, and Neiman. One noteworthy consequence of our proof is that it can be naturally generalized to obtain embeddability obstructions into non-positively curved spaces or asymptotically uniformly convex Banach spaces.

• Héctor Jardón-Sánchez (Leipzig University)

April 11, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title:  Measured property (T) and approximability"

Abstract: The aim of this talk is to present the following characterization: a p.m.p. countable borel equivalence relation (cber) has measured property (T) if and only if every ergodic extension of it is non-approximable. The starting point of the talk will be a dynamical characterization of measured property (T) à la Connes-Weiss, developed in joint work with Łukasz Grabowski and Sam Mellick. We will then study the dynamics induced by approximations of an equivalence relation and use our observations to prove the announced characterization. The remainder of the talk will be devoted to motivate said result through the discussion of some illustrative examples. Among these, we will see a brief proof of why graphings with planar connected components cannot have measured property (T).

• Simon Machado (ETH Zurich)

April 4, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title:  Doubling and Brunn—Minkowski in compact Lie groups

Abstract: Given a subset A of a locally compact group, the doubling constant is the ratio of the measure m(A^2) of the set A^2 of products of two elements of A to the measure m(A) of A. This constant is a central object in additive combinatorics, in the study of random walks on groups, in geometric analysis and in many other fields. 

In Euclidean spaces, doubling is now particularly well understood. Beyond that, the situation is far more mysterious. A conjecture of Breuillard and Green predicts that in a compact simple group this constant must be at least 2 to the power of the minimal co-dimension of a proper subgroup. 

In this talk, I will discuss the proof of this conjecture. I'll also explain how the tools employed open the door to other results, such as a Brunn-Minkowski inequality or a stability result.

• Aritra Ghosh (Renyi, Budapest)

March 21, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: On the General Rankin-Selberg Problem

Abstract:  We will see the general Rankin-Selberg problem. In a joint work, with Kummari Mallesham, Prof. Ritabrata Munshi, Saurabh Kumar Singh, we break the general Rankin-Selberg's bound (also the well known Rankin-Selberg's bound) on the error term. Our work will also generalize Bingrong Huang's result ``On the Rankin-Selberg problem" (Math. Ann. https://doi.org/10.1007/s00208-021-02186-7) and gives a better bound. This is an upcoming work.

• Carsten Peterson (Paderborn University)

March 7, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Quantum ergodicity on the Bruhat-Tits building for PGL(3) in the Benjamini-Schramm limit

Abstract: Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which "converge" to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.

• Petr Naryshkin (University of Münster)

February 29, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Topological tilings, mean dimension, and shift embeddability.

Abstract: We introduce the question of cubic shift embeddability for topological dynamical systems and explain the previously known results. We state our two main theorems:

1. systems with Uniform Rokhlin Property and comparison satisfy sharp shift embeddability and

2. for a large class of amenable groups, having a technical condition which we call property FCSB implies that the system has the Uniform Rokhlin Property and comparison. 

If time permits, we will attempt to sketch the proof of the first result.

• Karol Duda (IMPAN, Warsaw)

December 15, 14:00 (Faculty of Mathematics and Computer Science, room 0009). (note the new time)

Title: Torsion subgroups of small cancellation groups

Abstract: We prove that torsion subgroups of groups defined by, C(6), C(4)-T(4) or C(3)-T(6) small cancellation presentations are finite. 

This follows from more general results about locally elliptic action on small cancellation complexes.

• Matthias Uschold (Universität Regensburg)

November 24 , 14:00 (Faculty of Mathematics and Computer Science, room 0009).  (note the new time)

Title: Torsion homology growth and cheap rebuilding of inner-amenable groups

Abstract: In the first half, I will give a short introduction on inner-amenable groups and (torsion) homology growth invariants. One tool for showing vanishing of these invariants is the cheap rebuilding property, which was recently introduced by Mikołaj Fraczyk together with Abért, Bergeron and Gaboriau. We prove that certain inner-amenable groups have this property, thus extending vanishing results that were already known for amenable groups.

• Sohail Farhangi (UAM, Poznań)

November 9, 16:00 (Faculty of Mathematics and Computer Science, room 0009).
Title: Koopman representations for positive definite functions. 

Abstract: It is a classical result of Raikov and Gelfand that if G is a locally compact second countable group and \phi:G-> C is a continuous positive definite function, then there exists a representation U of G on Hilbert space H and a cyclic vector f\in H for which \phi(g)=<U(g)f,f>. Through the use of the Gaussian Measure Space Construction (GMSC), the previous result can be refined by taking H=L^2(X,\mu) and letting U be the Koopman representation of a measure preserving action of G. Our first main result is to further refine the latter result by showing that the measure preserving action of G can be assumed to be ergodic. Our second main result is when G is abelian, in which case we refine the result of the GMSC in a direction by showing that f\in L^2(X,\mu) can be taken to satisfy |f|=1 a.e. We will also review a classical result of Foias and Stratila that shows that we cannot always take the system in the previous result to be ergodic. If time permits, we will discuss connections to the study of van der Corput sets. 

Slides 

• Sami Douba (IHES, Bures-sur-Yvette)

October 26, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Quasi-arithmeticity and commensurability

Abstract: In their construction of nonarithmetic real hyperbolic lattices in each dimension, Gromov and Piatetski-Shapiro exploited the fact that two arithmetic lattices coinciding on a Zariski-dense subgroup must be commensurable. We discuss a rephrasing of the latter that holds more generally for lattices that are quasi-arithmetic in the sense of Vinberg, and how this can be used to demonstrate the abundance of compact hyperbolic Coxeter polyhedra in dimension 4, as well as of high-dimensional closed hyperbolic manifolds with equal small systole. This is largely based on joint work with Nikolay Bogachev and Jean Raimbault.

• Anne-Marie Aubert (IMJ-PRG, Paris)

October 19, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Application of extended quotients to the Langlands program

Abstract: We will first introduce the notion of extended quotient, which originates in noncommutative geometry. Next, we will explain how it naturally occurs in both the representation theory of p-adic reductive groups and the study of enhanced versions of local Langlands parameters. Finally, we will show how to interpret the local Langlands correspondence into a correspondence between extended quotients, and provide applications

The seminar will be preceded by an introductory lecture on representations of p-adic groups by Anna Szumowicz (IMPAN) 11:00-12:00 in the room 0146.