Thursdays 3:00-4:00pm in SRZ 216/217.
Annette Karrer (Technion), 14.10.21
Giles Gardam (Münster), 28.10.21
Ursula Hamenstädt (Bonn), 11.11.21
Xiaolei Wu (Shanghai Center for Mathematical Sciences), 18.11.21
Simon André (Münster), 25.11.21
Martin Nitsche (Karlsruhe), 9.12.21
Online via zoom (password protected).
Alice Kerr (Oxford), 15.4.21
Quasi-trees and product set growth
A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely studied notion of uniform exponential growth. We will see how quasi-trees could help us answer this question for acylindrically hyperbolic groups, and give a particular application to right-angled Artin groups.
Elia Fioravanti (MPI Bonn), 22.4.21
Coarse-median preserving automorphisms
We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If \phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix(\phi) is finitely generated and undistorted. Up to replacing \phi with a power, we show that Fix(\phi) is even quasi-convex with respect to the standard word metric. This implies that Fix(\phi) is separable and a special group in the Haglund-Wise sense. Some of our techniques are applicable in the more general context of Bowditch's coarse median groups. Based on arXiv:2101.04415.
Gareth Wilkes (Cambridge), 29.4.21
Residual properties of graphs of p-groups
When groups may be built up as graphs of ‘simpler’ groups, it is often of interest to study how good residual finiteness properties of simpler groups can imply residual properties of the whole. The essential case of this theory is the study of residual properties of graphs of finite groups. In this talk I will discuss the question of when a graph of finite p-groups is residually p-finite, for p a prime. I will describe the previous theorems in this area for one-edge and finite graphs of groups, and their method of proof. I will then state my recent generalisation of these theorems to potentially infinite graphs of groups, together with an alternative and more natural method of proof. Finally I will briefly describe a usage of these results in the study of accessibility—namely the existence of a finitely generated inaccessible group which is residually p-finite.
Çağrı Sert (Universität Zürich), 6.5.21
Large deviations and concentrations for random walks on hyperbolic spaces
In a first part, we will discuss large deviation phenomena and existence of large deviation principle for the displacement of random walks on hyperbolic spaces after having introduced related notions from probability theory. In a second part, we will focus on finite time bounds and concentrations for deviations of the displacement off the drift and mention an application to finite-time probabilistic Tits’ alternative. Based on joint works with Adrien Boulanger, Pierre Mathieu and Alessandro Sisto (part one above), and with Richard Aoun (part two above).
Maria Beatrice Pozzetti (Heidelberg), 20.5.21
Orbit growth rate in Higher rank Teichmüller spaces
For some non-compact semisimple Lie groups G there are connected components of the character variety Hom(\pi_1(S),G)/G that only consist of conjugacy classes of injective homomorphisms with discrete image, the higher rank Teichmüller spaces (HRTS). After introducing and motivating the study of HRTS, I will discuss joint work with Sambarino and Wienhard in which we study how the orbit growth rate of the associated actions on the Riemannian symmetric space G/K varies on the different HRTS.
Lvzhou Chen (UT Austin), 10.6.21
Stable torsion length
Many interesting groups are generated by torsion elements, for instance, mapping class groups, SL(n,Z) and Homeo^+(S^1). The word length with respect to this typically infinite generating set is called the torsion length. That is, the torsion length tl(g) of an element g is the smallest k such that g is the product of k torsion elements. The stable torsion length stl(g) is the limit of tl(g^n)/n, which measures the growth of the torsion length. I will explain how to use topological methods (and planar surfaces) to compute stl(g) in free products of finite abelian groups. The nature of the method implies that stl(g) is always rational in these free products. This is joint work with Chloe Avery.
Shi Wang (Michigan State University), 17.6.21
Kleinian groups of small critical exponent
Given a finitely generated, non-elementary discrete subgroup G < Isom(H^n), the orbit points grow exponentially with respect to the hyperbolic distance, and the critical exponent of G is defined to be the exponential growth rate. In this talk, I will present recent joint work with Beibei Liu. We show that if the critical exponent is small enough, then G is convex-cocompact, that is, the orbit map is a quasi-isometric embedding. Along the way, I'll also explain how a small critical exponent affects the geometry and topology of the quotient space.
Agatha Atkarskaya (Jerusalem), 24.6.21
Combinatorial approach for Burnside groups of relatively small odd exponents
In order to have a better understanding of our mathematical world, it seems to be of a big interest to study algebraic systems with unusual and even counter-intuitive properties. We consider a free group in a variety of groups with the identity $w^n = 1$. Namely, we study the group $$ B(m, n) = \langle x_1, \ldots, x_m \mid w^n = 1, w\in \langle x_1, \ldots, x_m\rangle \ldots \rangle, $$ it is called a free Burnside group of exponent~$n$. That is, the order of all elements in $B(n, m)$ is bounded by~$n$ which does not depend on a particular element. The question whether $B(n, m)$ is finite or infinite is in the spirit of such questions about objects with exotic properties and is called the general Burnside problem (stated by Burnside in 1902). The first solution of this problem was obtained in 1972 by Novikov and Adian for odd $n\geqslant 4381$, and then in 1982 by Olshanskii using different methods. We study $B(n, m)$ for $m \geqslant 2$ and odd numbers $n$ and our method gives a clear intuition why this group is infinite for big enough $n$. I will tell which effects make $B(n, m)$ to be infinite for odd $n \geqslant 297$. Joint work with Professor Eliyahu Rips and Professor Katrin Tent.
Kasia Jankiewicz (University of Chicago), 1.7.21
Boundary rigidity for groups acting on product of trees
The visual boundary is a well-defined compactification of a hyperbolic or CAT(0) space. For hyperbolic groups the boundary is unique up to homeomorphism. However, Croke-Kleiner constructed examples of CAT(0) groups acting geometrically on CAT(0) spaces with non-homeomorphic boundaries. I will discuss the question of the uniqueness of the boundary for groups acting geometrically on product of two trees. This is a wide family of groups including products of free groups, as well as some simple groups. This is joint work with Annette Karrer, Kim Ruane and Bakul Sathaye.
Bruno Martelli (Università di Pisa), 8.7.21
Hyperbolic 5-manifolds that fiber over the circle
We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions n<=5, a fact that leads us naturally to ask whether this may hold for any n. One consequence of this result is the existence of hyperbolic groups with finite-type subgroups that are not hyperbolic. The main tool is Bestvina - Brady theory applied to some hyperbolic n-manifolds that decompose very nicely into right-angled polytopes, enriched with the combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.
Stefan Witzel (Gießen), 15.7.21
Arithmetic approximate lattices and their finiteness properties
Approximate groups were identified as a natural framework for geometric group theory by Björklund and Hartnick and further developed by Cordes, Hartnick and Tonić, unifying previous research on apparently disparate areas such as finite approximate groups (Breuillard, Green, Tao) and quasi-crystals (Meyer and others). Approximate groups arise naturally via a cut-and-project procedure from lattices in locally compact groups. A central point I want to make is that S-arithmetic groups are, by their standard definition, the result of cut-and-project procedure. They happen to be groups as long as S contains all infinite places, an assumption usually imposed. In the context of approximate groups, that assumption can be lifted and gives rise to S-arithmetic approximate groups in characteristic 0 that are not groups but resemble S-arithmetic groups in positive characteristic. The finiteness properties of S-arithmetic subgroups of reductive groups in positive characteristic are determined by the Rank Theorem (joint with Bux and Köhl). I will present joint work with Tobias Hartnick proving a Rank Theorem for S-arithmetic approximate groups in characteristic 0.
Mondays 4:30-5:30pm, tea from 4:15pm
Robert Kropholler (Münster), 2.11.20
Algebraic fibering and incoherence for surface bundles
In this talk I will focus on the class of surface bundles over a surface. Such groups are a natural extension of fibered 3-manifolds. We are interested in what properties surface-by-surface groups have in common with fibered 3-manifold groups. Of particular interest is the question of fibering and incoherence. I will discuss the relations between these properties and use this to show that a such a bundle has coherent fundamental group if and only if the base is a torus. I will also discuss obstructions to such bundles' algebraic fibering. This is joint work with Stefano Vidussi and Genevieve Walsh.
Sam Shepherd (Oxford), 9.11.20
Quasi-isometric rigidity of generic cyclic HNN extensions of free groups
Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.
Bakul Sathaye (Münster), 16.11.20
Obstructions to Riemannian smoothings of locally CAT(0) manifolds
In this talk I will discuss obstructions to having a Riemannian metric with non-positive sectional curvature on a locally CAT(0) manifold. I will focus on the obstruction in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their method can be extended to construct new examples of locally CAT(0) 4-manifolds M that do not have a Riemannian smoothing. The universal covers of these manifolds satisfy the isolated flats condition and contain a collection of 2-dimensional flats with the property that their boundaries at infinity form non-trivial links in the boundary 3-sphere.
Federico Vigolo (Münster), 23.11.20
Coarse groups and their coarse actions
Much of geometric group theory can be restated in more general terms using the language of coarse spaces and coarse geometry. This is a very natural language that has been largely overlooked: this talk aims to remedy this situation. I will give a gentle introduction to the notions of (metric) coarse groups and their coarse actions. Rather than going for hard results, I will put emphasis on concrete examples and connections with classical notions and results. This will exemplify the conceptual clarity of the categorical/coarse geometric approach and point to various natural questions.
Jean Pierre Mutanguha (MPI Bonn), 30.11.20
Finding relative immersions of free groups
The overarching goal of the train track theory for free group automorphisms is finding the "best" ways to represent an automorphism so as to read off its dynamical properties. In this talk I will describe the progress I made in developing the theory for injective endomorphisms. To some degree, it turns out nonsurjective endomorphisms have simpler dynamics -- a result that I found surprising. If time permits, I may mention how the result was used to characterize when ascending HNN extensions of free groups are word-hyperbolic.
Nicolaus Heuer (Cambridge), 7.12.20
The Spectrum of Simplicial Volume
Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications. In dimensions two and three, the set of simplicial volumes of orientable closed connected (occ) manifolds may be fully described using geometrization. Much less is known in higher dimensions. In joint work with Clara Löh (University of Regensburg), we show that the set of simplicial volumes in dimensions four and higher is dense in the non-negative reals. I will also motivate a conjecture on the precise set of simplicial volumes in dimension 4.
Romain Tessera (Jussieu-Paris), 14.12.20
Dehn functions and large-scale geometry of nilpotent Lie groups
A long standing conjecture says that two simply connected nilpotent Lie groups are quasi-isometric if and only if they are isomorphic. A seminal result of Pansu reduces the problem of pairs of such groups with isomorphic Carnot-gradable Lie groups. In a joint work with Claudio Llosa Isenrich and Gabriel Pallier, we show that such pairs can have different Dehn functions. We deduce that the Dehn function is not an invariant of the asymptotic cone. As an application we answer a question of Cornulier on a natural variant of the above conjecture.
Matthew Cordes (ETH Zürich), 25.1.21
Geometric approximate group theory
An approximate group is a group that is "almost closed" under multiplication. Finite approximate subgroups play a major role in additive combinatorics. Recently Breuillard, Green and Tao have established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov's polynomial growth theorem. Infinite approximate groups were studied implicitly long before the formal definition. Approximate subgroups of R^n that are Delone sets can be constructed using "cut-and-project" methods and are models for mathematical quasi-crystals. Recently, Björklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and their geometric aspects. This is joint work with Hartnick and Tonic.
Calum Ashcroft (Cambridge), 1.2.21
Cubulating groups acting on polygonal complexes
Given a group G acting on a CAT(0) polygonal complex, X, it is natural to ask whether the structure of X allows us to deduce properties of G, for example whether G has a codimension-1 subgroup (and so does not have Property (T)), the Haageruup property, or even if G is special. We discuss some recent work on local properties that X may possess which allow us to answer these questions.
See page for 2019.