# Topological and Homological Methods in Group Theory 2022

## General Information

This is the third instalment of the THGT conference series. As its predecessors, the conference takes place in Bielefeld in spring, and as before it welcomes researchers studying groups through their interactions with topology and homology, as well as people interested in entering the subject.

If you have questions, please contact the organisers (Kai-Uwe Bux, Dawid Kielak and Stefan Witzel).

This conference is supported by the DFG priority programme 2026 Geometry at Infinity

## Invited Speakers

• Arthur Bartels
• Ursula Hamenstädt
• Tobias Hartnick
• Robert Kropholler
• Clara Löh
• Kim Ruane
• Alain Valette

## Programme

The talks on Monday and Thursday morning take place in the lecture hall H12 in the main building of the Bielefeld University.

The talks on Tuesday and Thursday afternoon take place in the lecture hall H5.

The talks on Wednesday and Friday take place in the lecture hall H13.

How do I find the university and the lecture hall?

For the hike we meet 14:00 in front of the Bielefelder Hof, which directly in front of the main station. The hike will be around 10 km long. There is the possibility to take a short cut (~7km).

The conference dinner will be at the Numa.

TimeTitleSpeaker
Monday, 28.03
09:30 - 10:30Dimensions for the mapping class groupUrsula Hamenstädt
10:30 - 11:00Coffee break
11:00 - 11:30A cubical Rips constructionMacarena Arenas
12:00 - 14:00Lunch
14:00 - 14:30High-dimensional rational cohomology of $SL_n(\mathbb{Z})$ and $Sp_{2n}(\mathbb{Z})$Benjamin Brück
14:30 - 14:45Short break
14:45 - 15:15Fibring of RFRS groups via L2 Betti numbersSam Fisher
15:15 - 15:45Coffee break
16:00 - 16:30Irreducible lattices fibring over the circleSam Hughes
Tuesday, 29.03
09:30 - 10:30Property T for some groups of tame automorphisms of $K[x,y,z]$Martin Kassabov
10:30 - 11:00Coffee break
11:00 - 12:00$K$-theory of Hecke algebrasArthur Bartels
12:00 - 14:00Lunch
14:00 - 14:30Compact uniformly recurrent subgroupsWaltraud Lederle
14:30 - 14:45Short break
14:45 - 15:15Stably free modules and the homotopy type of a finite $2$-complexJohn Nicholson
15:15 - 16:00Coffee break
16:00 - 16:30Stallings-Swan Theorem for unimodular totally disconnected locally compact groupsIlaria Castellano
Wednesday, 30.03
09:30 - 10:30Computations in bounded cohomologyClara Löh
10:30 - 11:00Coffee break
11:00 - 12:00Explicit Baum-Connes for semi-direct products of $\mathbb{Z}^2$ by some non-amenable subgroups of GAlain Valette
12:00 - 14:00Lunch
14:00 - 18:00Hike
19:00 - 23:59 (Conference dinner)
Thursday, 31.03
09:30 - 10:00An invitation to coarse groupsFederico Vigolo
10:30 - 11:00Coffee break
11:00 - 12:00Homological filling functions do not detect solubility of the word problem. Robert Kropholler
12:00 - 14:00Lunch
14:00 - 14:30Coarse Embeddings of Symmetric Spaces and Euclidean BuildingsOussama Bensaid
14:30 - 14:45Short break
14:45 - 15:15Obstructions to Riemannian smoothing of non-positively curved manifoldsBakul Sathaye
15:15 - 16:00Coffee break
16:00 - 17:00CAT(0) Groups; Example and Non-Examples (Kovalevskaya lecture)Kim Ruane
Friday, 01.04
09:30 - 10:30Random homotopies and applications to stability of bounded cohomology of arithmetic groupsTobias Hartnick
10:30 - 11:00Coffee break
11:00 - 11:30A bootstrapping argument for bounded-cohomological stability in degree three Carlos De la Cruz Mengual

## Abstracts

Macarena Arenas

#### A cubical Rips construction

The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group $Q$ and producing as an output a hyperbolic group $G$ that maps onto $Q$ with finitely generated kernel. The output group" $G$ is crafted by adding generators and relations to a presentation of $Q$, in such a way that these relations create enough noise" in the presentation to ensure hyperbolicity. One can then lift pathological properties of $Q$ to (some subgroup of) $G$. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem. In this talk, I will explain Rips’ result, mention some of its variations, and survey some tools and concepts related to these constructions, including small cancellation theory, cubulated groups, and asphericity. Time permitting, I will describe a variation of the Rips construction that produces cubulated hyperbolic groups of any desired cohomological dimension.
Arthur Bartels

#### $K$-theory of Hecke algebras

Hecke algebras of totally disconnected groups are natural analogs of group rings of discrete groups. Modules over Hecke algebras correspond to smooth representations of totally disconnected groups. The $K$-theory of Hecke algebra can be approached via a variant of the Farrell-Jones conjecture for them. I will give an introduction to this conjecture and explain similarities and difference to the discrete case. I will also discuss the proof of the conjecture for $p$-adic reductive groups.
Oussama Bensaid

#### Coarse Embeddings of Symmetric Spaces and Euclidean Buildings

Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. We will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. We show that, like in the quasi-isometric case, the rank is monotonous under coarse embeddings, provided that there is no Euclidean factor in the domain, or a Euclidean factor of dimension $1$. The proof involves higher homological filling functions.
Benjamin Brück

#### High-dimensional rational cohomology of $SL_n(\mathbb{Z})$ and $Sp_{2n}(\mathbb{Z})$

By a result of Church-Putman, the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in "codimension one", i.e. $H^{{n \choose 2} -1}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 3$, where ${n \choose 2}$ is the virtual cohomological dimension of $\operatorname{SL}_n(\mathbb{Z})$. I will talk about work in progress on two generalisations of this result: The first project is joint work with Miller-Patzt-Sroka-Wilson. We show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in codimension two, i.e. $H^{{n \choose 2} -2}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 4$. The second project is joint with Patzt-Sroka. Its aim is to study whether the rational cohomology of the symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes in codimension one, i.e. whether $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 2$.
Ilaria Castellano

#### Stallings-Swan Theorem for unimodular totally disconnected locally compact groups

The classical Stallings–Swan theorem states that a (discrete) group $G$ is free if, and only if, its cohomological dimension is less or equal to 1. It was firstly proved by J.R. Stallings for finitely generated groups, and subsequently extended to the general case by R.G. Swan. This result is considered as a milestone in the cohomology theory of discrete groups. Some time later, M. Dunwoody extended the Stallings–Swan theorem to all commutative rings. In this talk I will show that an analogue of the Stallings–Swan theorem holds in the realm of totally disconnected locally compact groups. Unfortunately, we were not able to generalize the theorem completely to this class of groups, but we had to assume further finiteness conditions, that are, compact generation and $CO$-boundedness. Based on a joint work with B. Marchionna and Th. Weigel, University of Milano Bicocca
Carlos De la Cruz Mengual

#### A bootstrapping argument for bounded-cohomological stability in degree three

In recent joint work with Tobias Hartnick, we have established the stabilization of bounded cohomology along a wide range of families of classical Lie groups indexed by the rank. A downside of our methods is that the obtained stability range does not allow to transport what is known for rank-one Lie groups to groups of higher rank in their corresponding families. In this talk, we will present an instance in which we are able to bootstrap any stability range into an optimal one. This, combined with our stability theorem, yields new computations of bounded cohomology of Lie groups in degree three.
Sam Fisher

#### Fibring of RFRS groups via L2 Betti numbers

A group is said to be fibred if it admits an epimorphism onto the integers with finitely generated kernel. We show that a RFRS group $G$ of type $FP_n(Q)$ virtually fibres with kernel of type $FP_n(Q)$ if and only if the first n $l^2$-Betti numbers of G vanish. As an application, we show that RFRS groups of type $FP(Q)$ with Noetherian group rings are polycyclic-by-finite, confirming a special case of a conjecture of Baer.
Ursula Hamenstädt

#### Dimensions for the mapping class group

The virtual cohomological dimension of the mapping class group of a closed surface $S$ of genus $g$ is known to be $4g-5$. Improving slightly a result of Gabai, with a different proof, we show that the covering dimension of the boundary of the curve graph is at most $4g-6$, and we give some evidence that equality holds. If time permits, we'll sketch a conjectural picture of the large scale topological and geometric structure of the mapping class group.
Tobias Hartnick

#### Random homotopies and applications to stability of bounded cohomology of arithmetic groups

Bounded cohomology of countable groups is a powerful tool capturing hyperbolic behaviour at infinity but notoriously hard to compute. Rather than studying the bounded cohomology of a single group it is often easier to consider bounded cohomology along families of groups. For classical families of arithmetic groups of increasing rank we can show that bounded cohomology stabilizes (against the same values as the continuous bounded cohomology of the family of ambient Lie groups). We explain some of the main ingredients in the proof: A theorem of Monod relating bounded cohomology of lattices to continuous bounded cohomology of their lattice envelopes, a version of Quillen’s stability criterion adapted to bounded cohomology, Stiefel complexes of classical groups (as first introduced by Vogtmann) and finally a new technique of establishing acyclicity in bounded cohomology through homotopies which involve random configurations of points. We will explain how random points in Stiefel complexes can achieve configurations which cannot be achieved by deterministic points and how this is useful in constructing homotopies. This is based on joint work with Carlos De la Cruz Mengual.
Sam Hughes

#### Irreducible lattices fibring over the circle

An application of the Margulis Normal Subgroup Theorem shows the following: Let $\Gamma$ be a lattice in a semi-simple Lie group with finite centre. If $H^1(\Gamma)$ is non-zero, then $\Gamma$ is reducible. In this talk we will investigate a natural generalisation: Let $\Lambda$ be a lattice in a product of $\mathrm{CAT}(0)$ spaces. If the BNSR invariant $\Sigma^m(\Lambda)$ is non-empty for some $m\geq1$, is $\Lambda$ reducible?
Martin Kassabov

#### Property T for some groups of tame automorphisms of $K[x,y,z]$

I will describe a construction of some subgroups of automorphisms of polynomial rings like $K[x,y,z]$ consisting of tame automorphisms which have Kazdan property T. The groups can be used to provide new examples of expander graphs. This is joint work with Pierre-Emmanuel Caprace.
Robert Kropholler

#### Homological filling functions do not detect solubility of the word problem.

For finitely generated groups the word problem asks for the existence of an algorithm that takes in words in a finite generating set and decides if a word is trivial or not. For finitely presented groups this is equivalent to the Dehn function being sub-recursive. There is an analogue of the Dehn function for groups of type $FP_2$, this function measures the difficulty of filling loops in a certain space with surfaces. In joint work with Noel Brady and Ignat Soroko, we give computations of the homological filling functions for Ian Leary's groups $G_L(S)$. We use this to show that there are uncountably many groups with homological filling function $n^4$. This gives groups that have sub-recursive homological filling function but unsolvable word problem.
Waltraud Lederle

#### Compact uniformly recurrent subgroups

Given a locally compact group, the set of closed subgroups carries a compact, Hausdorff topology and is commonly referred to as the "Chabauty space" of the group. A URS is a minimal, conjugation-invariant, closed subset of the Chabauty space; it is thought of as dynamical counterpart to the more well-known IRS (invariant random subgroup). We prove that the union of a URS that consists solely of compact subgroups has to be contained in a compact normal subgroup. This is joint work with Pierre-Emmanuel Caprace and Gil Goffer.
Clara Löh

#### Computations in bounded cohomology

Bounded cohomology of spaces or groups is the cohomology of the bounded dual of the singular chain complex or the simplicial resolution, respectively. Bounded cohomology has various applications in geometric topology and group theory. In contrast to ordinary group cohomology, much of the overall picture of bounded cohomology remains unclear. In this talk, I will report on recent progress in computations of bounded cohomology. This is based on joint work with Francesco Fournier-Facio and Marco Moraschini.
John Nicholson

#### Stably free modules and the homotopy type of a finite $2$-complex

Two presentations for a group $G$ which have the same deficiency are called exotic if the corresponding presentation $2$-complexes are not homotopy equivalent. The first examples of exotic presentations were found by Dunwoody and Metzler in the 1970s. A long-standing problem, which is Problem D5 in the 1979 Problems List of C. T. C. Wall, is to determine whether there exist exotic presentations with non-minimal deficiency. For many groups $G$, there is a close relationship between the homotopy types of group presentations for $G$ and stably free ZG-modules, with deficiency corresponding to the rank of the stably free module. Dunwoody’s exotic presentations were constructed using non-free stably free modules of rank 1 over the Trefoil group. However, no examples of non-free stably free $ZG$-modules of rank greater than 1 have since been found. The aim of this talk will be to present the first examples in each case. In particular, for each $k$ at least $2$, I will construct non-free stably free ZG-modules of rank $k$ and exotic presentations with deficiency $k$ above the minimal value. I will also present a number of new questions.
Kim Ruane

#### CAT(0) Groups; Example and Non-Examples (Kovalevskaya lecture)

In this talk, we will discuss the class of CAT(0) groups. A finitely generated group G is a CAT(0) group if G acts properly discontinuously and cocompactly by isometries on a CAT(0) space. We will recall several well-known examples as well as non-examples in the hopes of understanding methods for constructing CAT(0) spaces as well as obstructions for showing that no such space can be found. We will also discuss groups for which the question of being CAT(0) is still open. It is a well-known open problem to decide whether the braid groups on more than 7 strands are CAT(0) or not. We will focus on a different family of examples - namely, the automorphism groups of right-angled Coxeter groups and their subgroups.
Bakul Sathaye

#### Obstructions to Riemannian smoothing of non-positively curved manifolds

In this talk I will discuss obstructions to having a smooth Riemannian metric with non-positive sectional curvature on a locally CAT(0) manifold. I will briefly discuss a knotting obstruction in dimension $= 4$ and explain why this obstruction does not work in higher dimensions. I will then discuss a large scale quasi-isometry invariant of pairs of spaces $(X,Y)$ called the 'fundamental group of the $Y$-end' and show how it can be used to give a coarse knotting obstruction to Riemannian smoothings in higher dimensions. This is joint work with Jean-Francois Lafont.
Alain Valette

#### Explicit Baum-Connes for semi-direct products of $\mathbb{Z}^2$ by some non-amenable subgroups of G

Semi-direct products $\mathbb{Z}^2\rtimes G$ (with $G$ non-amenable) are interesting because they satisfy the Baum-Connes conjecture without belonging to one of the large classes (a-T-menable groups and hyperbolic groups) for which the conjecture is known to hold. Thanks to a 3-dimensional model for the classifying space of proper actions of $\mathbb{Z}^2\rtimes G$, the geometric side of the conjecture can be explicitly computed, and so gives intuition for the analytical side. In good cases a proof by hand of the Baum-Connes conjecture can be obtained: we will give two examples. This is part of a joint project with R. Flores, S. Pooya and A. Zumbrunnen.
Federico Vigolo

#### An invitation to coarse groups

Informally, coarse groups are spaces equipped with operations that satisfy the group axioms up to uniformly bounded error. Formally, they are group objects in the category of coarse spaces. Besides being an interesting subject in its own right, the study of coarse groups and their coarse homomorphisms has connections with a number of different topics going from geometric group theory to number theory and functional analysis. This talk is a brief introduction and survey of the subject, with emphasis on examples and connections.
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## Participants

• Abels, Herbert (Universität Bielefeld)
• Andritsch, Konstantin (ETH Zürich)
• Arenas, Macarena (University of Cambridge)
• Bargagnati , Giuseppe (Università di Pisa )
• Baumeister, Barbara (University Bielefeld)
• Bensaid, Oussama (Université de Paris)
• Bhattacherjee, Sandipan (Birla Institute of Technology Mesra, India)
• Blumer, Simone (Università degli Studi di Milano-Bicocca)
• Bonn, Laura (KIT - Karlsruher Institut für Technoligie)
• Brück, Benjamin (ETH Zürich)
• Camacho Cadena, Fernando (RKU Heidelberg)
• Capovilla, Pietro (SNS Pisa)
• Cassella, Alberto (University of Milano - Bicocca / University of Zaragoza)
• Castellano, Ilaria (University of Milano Bicocca)
• Chanfi, Dorian (Justus-Liebig-Universität Gießen)
• Davies, Isobel (Otto-von-Guericke-Universität Magdeburg)
• De la Cruz Mengual, Carlos (Weizmann Institute of Science)
• Escalier, Amandine (WWU Münster)
• Fisher, Sam (University of Oxford)
• Flechsig, Jonas (Universität Bielefeld)
• Fournier-Facio, Francesco (ETH Zurich)
• Friedl, Stefan (Universität Regensburg)
• Fruhling Dutra, Ederson Ricardo (University of Regensburg / University of Sao Carlos (Brazil))
• García Mejía, Jerónimo (Karlsruher Institut für Technologie)
• Gardam, Giles (University of Münster)
• Gerasimova , Maria (WWU Münster)
• Hughes, Sam (University of Oxford)
• Incerti-Medici, Merlin (IHES)
• Kassabov, Martin (Cornell University)
• Kielak, Dawid (University of Oxford)
• Kirschner, Lovis (Justus-Liebig-Universität Gießen)
• Klinge, Kevin (Karlsruhe Intitute for Technology)
• Kudlinska, Monika (University of Oxford)
• Le Bars, Corentin (Université Paris Saclay)
• Lederle, Waltraud (UCLouvain)
• Leoni, Giorgio (Università degli Studi Milano-Bicocca)
• Li, Kevin (University of Southampton)
• Linton, Marco (University of Warwick)
• Lopez de Gamiz Zearra, Jone (University of Warwick)
• López Neumann, Antonio (Ecole polytechnique)
• Marasco, Domenico (University of Pisa)
• Marchionna, Bianca (University of Milano-Bicocca)
• Martínez Pérez, Conchita (Universidad de Zaragoza)
• Messaci, Mohamed Lamine (Université Côte d'Azur)
• Michael, Anna Franziska (Otto-von-Guericke Universität Magdeburg)
• Milizia, Francesco (SNS Pisa)
• Mineh, Lawk (University of Southampton)
• Mizerka, Piotr (Institute of Mathematics, Polish Academy of Sciences)
• Mohammadiyeh, Seyyed Ali (University of Kashan)
• Munser, Lars (Universität Regensburg)
• Nicholson, John (Imperial College London)
• Niemeyer, Merik (RKU Heidelberg)
• Nitsche, Martin (Karlsruhe Institute of Technology)
• Puder, Doron (Tel Aviv University)
• Quintanilha, José Pedro (Universität Bielefeld)
• Saleh, Nada (Justus Liebig Universität)
• Santos Rego, Yuri (Otto-von-Guericke Universität Magdeburg)
• Sathaye, Bakul (WWU Münster)
• Schesler, Eduard (FernUni Hagen)
• Schwer, Petra (OVGU Magdeburg)
• Seo, Donggyun (Seoul National University)
• Souza, Henrique Augusto (Universidad Autónoma de Madrid)
• Sun, Bin (University of Oxford)
• Thilmany, François (KU Leuven)
• Titz Mite, Thomas (JLU Gießen)
• Uschold, Matthias (Universität Regensburg)
• Varghese, Olga (OVGU Magdeburg)
• Vigolo, Federico (Münster University)
• Weigel, Thomas (University of Milano-Bicocca)
• Witzel, Stefan (JLU Gießen)
• Zumbrunnen, Alexandre (Université de Neuchâtel)