Forthcoming events in this series


Tue, 21 May 2024
15:00
L5

Rigidity and automorphisms of group von Neumann algebras

Denis Osin
Abstract

I will survey some recent results on rigidity and automorphisms of von Neumann algebras of groups with Kazhdan property (T) obtained in a series of joint papers with I. Chifan, A. Ioana, and B. Sun. Specifically, we show that certain groups, constructed via a group-theoretic version of Dehn filling in 3-manifolds, satisfy several conjectures proposed by A. Connes, V. Jones, and S. Popa. Previously, no nontrivial examples of groups satisfying these conjectures were known. At the core of our approach is the new notion of a wreath-like product of groups, which seems to be of independent interest.

Tue, 14 May 2024
15:00
L6

Extension of Möbius boundary homeomorphisms

Urs Lang
Abstract
In this talk, I will review recent results of K. Biswas. It is an open problem whether 
every Möbius homeomorphism between the visual boundaries of two Hadamard 
manifolds of curvature at most -1 extends to an isometry between them. A positive 
answer would resolve the long-standing marked length spectrum rigidity conjecture 
of Burns-Katok for closed negatively curved manifolds. Biswas' results yield an 
isometry between certain functorial thickenings of the manifolds, which lie within 
uniformly bounded distance and can be identified with their injective hulls.
Tue, 07 May 2024
15:00
L6

Oka manifolds and their role in complex analysis and geometry

Franc Forstneric
Abstract

Oka theory is about the validity of the h-principle in complex analysis and geometry. In this expository lecture, I will trace its main developments, from the classical results of Kiyoshi Oka (1939) and Hans Grauert (1958), through the seminal work of Mikhail Gromov (1989), to the introduction of Oka manifolds (2009) and the present state of knowledge. The lecture does not assume any prior exposure to this theory.

Tue, 30 Apr 2024
15:00
L6

Graph products and measure equivalence

Camille Horbez
Abstract

Measure equivalence was introduced by Gromov as a measure-theoretic analogue to quasi-isometry between finitely generated groups. In this talk I will present measure equivalence classification results for right-angled Artin groups, and more generally graph products. This is based on joint works with Jingyin Huang and with Amandine Escalier. 

Tue, 23 Apr 2024
15:00
L6

Approximate lattices: structure and beyond

Simon Machado
Abstract

Approximate lattices are aperiodic generalisations of lattices in locally compact groups. They were first introduced in abelian groups by Yves Meyer before being studied as mathematical models for quasi-crystals. Since then their structure has been thoroughly investigated in both abelian and non-abelian settings.

In this talk I will survey what is known of the structure of approximate lattices. I will highlight some objects - such as a notion of cohomology sitting between group cohomology and bounded cohomology - that appear in their study. I will also formulate open problems and conjectures related to approximate lattices. 

Tue, 05 Mar 2024
15:00
L6

Sharp spectral gaps for scl from negative curvature

Alexis Marchand
Abstract

Stable commutator length is a measure of homological complexity of group elements, which is known to take large values in the presence of various notions of negative curvature. We will present a new geometric proof of a theorem of Heuer on sharp lower bounds for scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl.

Tue, 27 Feb 2024
15:00
L6

Sublinear rigidity of lattices in semisimple Lie groups

Ido Grayevsky
Abstract

I will talk about the coarse geometry of lattices in real semisimple Lie groups. One great result from the 1990s is the quasi-isometric rigidity of these lattices: any group that is quasi-isometric to such a lattice must be, up to some minor adjustments, isomorphic to lattice in the same Lie group. I present a partial generalization of this result to the setting of Sublinear Bilipschitz Equivalences (SBE): these are maps that generalize quasi-isometries in some 'sublinear' fashion.

Tue, 20 Feb 2024
15:00
L6

Counting geodesics of given commutator length

Viveka Erlandsson
Abstract

Abstract: It’s a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface S is asymptotic to exp(L)/L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. One such result is the asymptotic growth of those that are homologically trivial, proved independently by both by Phillips-Sarnak and Katsura-Sunada. A homologically trivial curve can be written as a product of commutators, and in this talk we will look at those that can be written as a product of g commutators (in a sense, those that bound a genus g subsurface) and obtain their asymptotic growth. As a special case, our methods give a geometric proof of Huber’s classical theorem. This is joint work with Juan Souto. 

Tue, 13 Feb 2024
15:00
L6

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Nansen Petrosyan
Abstract

We introduce a new class of groups with Thompson-like group properties. In the surface case, the asymptotic mapping class group contains mapping class groups of finite type surfaces with boundary. In dimension three, it contains automorphism groups of all finite rank free groups. I will explain how asymptotic mapping class groups act on a CAT(0) cube complex which allows us to show that they are of type F_infinity. 

This is joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Xaolei Wu.

Tue, 06 Feb 2024
15:00
L6

On the abelianization of the level 2 congruence group of the mapping class group.

Tara Brendle
Abstract

We will survey work of Birman-Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

Tue, 30 Jan 2024
15:00

How hard is it to know if there is an epimorphism from one group to another

Murray Elder
Abstract

Let C,D be classes of finitely presented groups. The epimorphism problem from C to D is the following decision problem:

Input: Finite descriptions (presentation, multiplication table, other) for groups  G in C and H in D

Question: Is there an epimorphism from G to H?

I will discuss some cases where it is decidable and where it is NP-complete. Spoiler alert: it is undecidable for C=D=the class of 2-step nilpotent groups (Remeslennikov).

This is joint work with Jerry Shen (UTS) and Armin Weiss (Stuttgart).

Tue, 23 Jan 2024
15:00
L6

Cocycle and orbit equivalence superrigidity for measure preserving actions

Daniel Drimbe
Abstract

The classification of measure preserving actions up to orbit equivalence has attracted a lot of interest in the last 25 years. The goal of this talk is to survey the major discoveries in the field, including Popa's cocycle and orbit equivalence superrigidity theorem and discuss some recent superrigidity results for dense subgroups of Lie groups acting by translation.

Tue, 16 Jan 2024
15:00
L6

Profinite invariants of fibered groups

Monika Kudlinska
Abstract

A central question in infinite group theory is to determine how much global information about a group is encoded in its set of finite quotients. In this talk, we will discuss this problem in the case of algebraically fibered groups, which naturally generalise fundamental groups of compact manifolds that fiber over the circle. The study of such groups exploits the relationships between the geometry of the classifying space, the dynamics of the monodromy map, and the algebra of the group, and as such draws from all of these areas.

Tue, 28 Nov 2023
15:00
L1

Fixed points of group homomorphisms and the Post Correspondence Problem

Laura Ciobanu
Abstract

The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms g and h between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. 

While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. In this talk I will give an overview of what is known about the PCP in hyperbolic groups, nilpotent groups and beyond (joint work with Alex Levine and Alan Logan).

Tue, 21 Nov 2023
15:00
L1

Residual finiteness and actions on trees

Gareth Wilkes
Abstract

One of the more common ways to study a residually finite group (or its profinite completion) is via breaking it down into a graph of groups in some way. The descriptions of this theory generally found in the literature are highly algebraic and difficult to digest. I will present alternative, more geometric, definitions and perspectives on these theories based on properties of virtually free groups and their profinite completions.

Tue, 14 Nov 2023
15:00
L1

Classifiability of crossed products by nonamenable groups

Julian Kranz
Abstract

The celebrated Kirchberg-Phillips classification theorem classifies so-called Kirchberg algebras by K-theory. Many examples of Kirchberg algebras can be constructed via the crossed product construction starting from a group action on a compact space. One might ask: When exactly does the crossed product construction produce a Kirchberg algebra? In joint work with Gardella, Geffen, and Naryshkin, we obtained a dynamical answer to this question for a large class of nonamenable groups which we call "groups with paradoxical towers". Our class includes many non-positively curved groups such as acylindrically hyperbolic groups and lattices in Lie groups. I will try to advertise our notion of paradoxical towers, outline how we use it, and pose some open questions.

Tue, 07 Nov 2023
15:00

From strong contraction to hyperbolicity

Stefanie Zbinden
Abstract

For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.

Tue, 31 Oct 2023
15:00

Coarse obstructions to cubulation

Harry Petyt
Abstract

Given a group $G$, finding a geometric action of $G$ on a CAT(0) cube complex can be used to say some rather strong things about $G$. Such actions are not always easy to find, however, which makes it useful to have sufficient conditions, both for existence and for non-existence. This talk concerns the latter: we shall see a coarse geometric obstruction to a group admitting a cocompact cubulation. Based on joint work with Zach Munro.

Tue, 24 Oct 2023
15:00

Measure doubling for small sets in SO(3,R).

Yifan Jing
Abstract

Let $SO(3,R)$ be the $3D$-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Confirming a conjecture by Breuillard and Green, we show that if $A$ is an open subset of $SO(3,R)$ with sufficiently small measure, then $\mu(A^2) > 3.99 \mu(A)$. This is joint work with Chieu-Minh Tran (NUS) and Ruixiang Zhang (Berkeley). 

Tue, 17 Oct 2023
15:00

Dehn functions of central products of nilpotent groups

Claudio Llosa Isenrich
(KIT)
Abstract

The Dehn function of a finitely presented group provides a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. By work of Gersten, Holt and Riley the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$. However, we are still far from determining the precise Dehn functions of all nilpotent groups. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every $k>2$, we obtain many pairs of finitely presented $k$-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups.  This talk is based on joint works with García-Mejía, Pallier and Tessera.

Tue, 10 Oct 2023
15:00
L1

Rank gradient in higher rank lattices

Mikołaj Frączyk
(Jagiellonian University Cracow)
Abstract

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest. 

Tue, 10 Oct 2023

14:00 - 15:00
L5

Residual finiteness growth functions of surface groups with respect to characteristic quotients

Mark Pengitore
(University of Virginia)
Abstract

Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.

Tue, 13 Jun 2023

15:00 - 16:00
L4

Surface subgroups, virtual homology and finite quotients

Jonathan Fruchter
Abstract

We begin with a seemingly simple question: how can one distinguish a surface group from other cyclic amalgamations of two free groups? This question will prompt a (geometrically flavoured) investigation of virtual homological properties of graphs of free groups amalgamated along cyclic edge groups, where surface subgroups play a key role. 

We next turn to study limit groups and residually free groups through their finite quotients, and apply our findings to the study of profinite rigidity within these classes of groups. In particular, we will sketch out why a direct product of free and surface groups cannot have the same finite quotients as any other finitely presented residually free group.

If time permits, we will discuss other possible characterizations of surface groups among limit groups. The talk is based on joint work with Ismael Morales.

 

Tue, 06 Jun 2023

15:00 - 16:00
L4

Generating tuples of Fuchsian groups

Richard Weidmann
Abstract

Generating n-tuples of a group G, or in other words epimorphisms Fₙ→G are usually studied up to the natural right action of Aut(Fₙ) on Epi(Fₙ,G); here Fₙ is the free group of n generators. The orbits are then called Nielsen classes. It is a classic result of Nielsen that for any n ≥ k there is exactly one Nielsen class of generating n-tuples of Fₖ. This result was generalized to surface groups by Louder.

In this talk the case of Fuchsian groups is discussed. It turns out that the situation is much more involved and interesting. While uniqueness does not hold in general one can show that each class is represented by some unique geometric object called an "almost orbifold covers". This can be thought of as a classification of Nielsen classes. This is joint work with Ederson Dutra.

Tue, 30 May 2023

15:00 - 16:00
L3

On fundamental groups of an affine manifolds

Gregory Soifer
Abstract

The study of the fundamental group of an affine manifold has a long history that goes back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact Euclidian affine manifold has a subgroup of a finite index such that every element of this subgroup is translation. The motivation was the study of the symmetry groups of crys- talline structures which are of fundamental importance in the science of crystallography. A natural way to generalize the classical problem is to broaden the class of allowed mo- tions and consider groups of affine transformations. In 1964, L. Auslander in his paper ”The structure of complete locally affine manifolds” stated the following conjecture, now known as the Auslander conjecture: The fundamental group of a compact complete locally flat affine manifold is virtually solvable.

In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”, J.Milnor asked if a free group can be the fundamental group of complete affine flat mani- fold.
The purpose of the talk is to recall the old and to talk about new results, methods and conjectures which are important in the light of these questions .

The talk is aimed at a wide audience and all notions will be explained 1