Perspectives on the cohomology of general linear groups

This workshop focuses on recent advances around the (co-)homology of general linear and related groups. These basic topological invariants are, for example, related to questions in algebraic K-theory and conjectures in number theory. Their study can be approached from many different mathematical perspectives, each using their own set of techniques. The goal of this event is to stimulate exchange between these different points of view, to bring together experts, as well as to introduce PhD students and Postdocs to this exciting area of research.

The core of the workshop is formed by several

• lecture series and
• invited talks.

These are supplemented by

• contributed talks (that participants can apply for during the registration process) and,
• a poster session (in which all participants are invited to present their work).

The poster session is scheduled for Monday evening, and will be combined with a reception. On Thursday evening, participants are invited to join us for a conference dinner.

List of speakers

Lecture series by:

• Philippe Elbaz-Vincent (Université Grenoble Alpes)
  Title: "Cohomology of arithmetic groups, Voronoi's complexes and K-theory"
• Sam Payne (University of Texas at Austin)
  Title: "Hopf structures in the cohomology of \(\operatorname{GL}_n(\mathbb{Z})\) and \(\operatorname{Sp}_{2n}(\mathbb{Z})\)"
• Dan Petersen (Stockholm University)
  Title: "Homological stability and moments in families of L-functions over function fields"

Invited talks by:

• Holger Kammeyer (Heinrich-Heine-Universität Düsseldorf)

• Jeremy Miller (Purdue University)

• Erik Panzer (University of Oxford)

• Nathalie Wahl (University of Copenhagen)

• Jennifer Wilson (University of Michigan)

Contributed talks by:

• Fabio Capovilla-Searle (Purdue University)

• Karol Janowicz (Warsaw Doctoral School of Mathematics and Computer Science)

• Anja Meyer (Loughborough University)

• Piotr Mizerka (Adam Mickiewicz University in Poznań)

• Vikram Nadig (Bielefeld University)

• Matthew Scalamandre (University of Toronto)

• Ismael Sierra (University of Toronto)

• Markus Szymik (University of Sheffield)

• Emiliano Torti (University of French Polynesia)

• Raluca Vlad (Brown University)

Schedule and Abstracts

Preliminary schedule:

	Monday	Tuesday	Wednesday	Thursday	Friday
8:30-9:00	Registration (ground floor Einsteinstr. 62)
9:00-10:00	Elbaz-Vincent 1	Petersen 2	Elbaz-Vincent 2	Petersen 3	Payne 3
10:00-10:30	Coffee break	Coffee break	Coffee break	Coffee break	Coffee break
10:30-11:30	Petersen 1	Payne 1	Payne 2	Elbaz-Vincent 3	Panzer
11:45-12:30	Meyer	Szymik	Mizerka	Nadig	Janowicz
12:30-14:00	Lunch break	Conference picture Lunch break	Lunch break	Lunch break	End of conference
14:00-15:00	Kammeyer	Wahl	MM Common Ground Room: SRZ 216/217 (quite room near coffee break)	Miller
15:00-15:30	Coffee break	Coffee break	Free afternoon	Coffee break
15:30-16:15	Sierra	Torti		Capovilla-Searle
16:15-17:00	Scalamandre	Wilson (ends at 17:15)		Vlad
	Reception and poster session			19:00: Conference dinner

Lecture Series:

• Philippe Elbaz-Vincent – Cohomology of arithmetic groups, Voronoi's complexes and K-theory
  The cohomology of arithmetic groups, and more generally of linear groups, is a rich subject with links to geometry, topology, algebra and number theory.  In this lecture series, I will give an overview on several explicit results on the cohomology of \(\operatorname{SL}_N(\mathbb{Z})\) and related groups, their homologies with coefficients in their Steinberg modules, computations of their geometric models (in particular based on the Voronoi complexes) and related conjectures. I will also give applications to moduli spaces of curves and K-theory of the integers. The talks are based in part on several joint works of the author.

• Sam Payne – Hopf structures in the cohomology of \(\operatorname{GL}_n(\mathbb{Z})\) and \(\operatorname{Sp}_{2n}(\mathbb{Z})\)
  I will construct and explain two related Hopf algebra structures, one on the union of the compactly supported cohomology groups of the locally symmetric spaces for \(\operatorname{GL}_n(\mathbb{Z})\) and the other on the union of the weight zero compactly supported cohomology groups of the locally symmetric spaces for \(\operatorname{Sp}_{2n}(\mathbb{Z})\). Both of these Hopf algebras are bigraded (by the cohomological degree and the genus n), connected, and graded co-commutative. In the symplectic case, the weight zero subspace refers to Deligne’s weight filtration induced by viewing the locally symmetric space for \(\operatorname{Sp}_{2n}(\mathbb{Z})\) as the algebraic moduli space of principally polarized abelian varieties of complex dimension n. The structure of these compactly supported cohomology groups are related, via Poincaré duality, to the corresponding cohomology groups with coefficients in the orientation sheaves of the locally symmetric spaces. These talks will be based on recent joint work with Brown, Chan, and Galatius (https://arxiv.org/abs/2405.11528).

• Dan Petersen – Homological stability and moments in families of L-functions over function fields
  This is a report of joint work with Bergström–Diaconu–Westerland and Miller–Patzt–Randal-Williams. The goal of our two papers is to prove a theorem in analytic number theory, using homological stability. The proof uses many diverse tools, some of which are close to the topic of the workshop (Borel's results on stable real cohomology of arithmetic groups, high connectivity of simplicial complexes, cellular \(E_k\)-algebras) and some of which are further afield (Segal-style scanning for moduli spaces, logarithmic étale cohomology, Koszul duality of operads, combinatorics of symmetric functions). I will give an overview of the proof, aimed at an audience with no background in number theory or algebraic geometry.

Research talks:

• Fabio Capovilla-Searle - Top degree cohomology of congruence subgroups of symplectic groups
  The cohomology of arithmetic groups has connections to many areas of mathematics such as number theory and diffeomorphism groups. Classifying spaces of congruence subgroups of symplectic groups have an algebro-geometric interpretation as the moduli space of principally polarized abelian varieties with level structures. These congruence subgroups Sp_2n(Z,L) are the kernel of the mod-L reduction map Sp_2n(Z) to Sp_2n(Z / L). By work of Borel-Serre, H^i(Sp_2n(Z / L)) vanishes for i > n^2. I will report on lower bounds in the top degree i = n^2. The key tools in the proof are the theory of Steinberg modules and highly connected simplicial complexes.
• Karol Janowicz - Cohomology of general linear group and functor categories
  Every functor on the category of finite-dimensional vector spaces (over fixed, finite field k) into the category of vector spaces induces a coherent family of GL_n(k)-representations. Likewise, every strict polynomial functor (in Friedlander-Suslin sense) induces a family of polynomial representations of GL_n and this observation turns out to be very fruitful in representation theory. That is due to the comparison theorems which allow us to pass from the cohomology of rational GL_n-modules to the cohomology of strict polynomial functors and also pass from the cohomology of strict polynomial functors to the cohomology of naive functors. The stable character of these results makes them work very well for sufficiently big n. During the talk, I will concentrate on the comparisons related to the functor categories with bounded domain and possible approaches to calculations of Ext groups of GL_n(k)-modules for small n.
• Holger Kammeyer - Homology growth and profinite rigidity of arithmetic groups
  We will explain that the asymptotic growth of the free and the torsion part of the homology of arithmetic groups along finite index subgroups should be governed by so called L2-invariants. As an application, we will discuss in how far certain invariants of arithmetic groups (L2-Betti numbers, Euler characteristic, and volume) are already determined by the profinite completion.
• Anja Meyer - The modular cohomology of SL_2(Z/3^n)
  The general question of the modular cohomology of SL_n(Z/p^n) for n>1 and prime p is wide open, and the same holds for GL_n. In this talk I will present recent results that facilitate the computation for n=2 using profinite arguments, spectral sequences, and fusion systems. I shall present the mod 3 cohomology of SL_(Z/3^n) as a worked example.
• Jeremy Miller - Duoidal Hopf algebras and congruence subgroups
  I will describe an exotic algebraic structure called "duoidal Hopf algebras" and some applications of it to the cohomology of congruence subgroups of special linear groups of number rings. This is joint work with Avner Ash and Peter Patzt.
• Piotr Mizerka - Non-vanishing of group cohomology of SL(n,Z) in the rank
  It has been recently shown by Bader and Sauer that the cohomology of SL(n,Z) with coefficients in orthogonal representations without non-trivial invariant vectors vanishes below the rank. We show that for n=3 and n=4 their result is sharp: we indicate such specific representations for which SL(n,Z) possesses non-trivial cohomology in the rank. We apply the Steinberg duality which allows us to compute the group cohomologies of interest by means of a specific model of a symmetric space. The key idea is to translate such computations to calculations stemming from group rings. The latter could be accomplished with the help of computers. This is the joint work with B. Brück, S. Hughes, and D. Kielak
• Vikram Nadig - Homological stability for odd orthogonal groups
  I will present a homological stability result for isometries of symmetric bilinear forms ("odd orthogonal groups") for a class of principal ideal domains that includes the integers and finite fields of characteristic 2. I will then explain the relevance of this result in the context of Grothendieck-Witt theory. This is work in progress.
• Erik Panzer - Cohomology of GL_2n from differential forms
  I will explain how invariant algebraic differential forms give rise to concrete dual cohomology classes of the general linear group, focusing in particular on the case of even rank. Since these classes are primitive in the Hopf algebra, their products provide plenty of non-zero classes, which fully explain the cohomology in low ranks. This is part of joint work with Francis Brown and Simone Hu (arXiv:2406.12734).
• Matthew Scalamandre - Steinberg modules for rings
  The Steinberg module of a field F is a classical representation of GL_n(F) which encodes fundamental information about that group. In this talk, I will define the Steinberg module functorially, for a general ring R. We will discuss various algebraic properties of these modules, and consider some applications to the cohomology of arithmetic groups.
• Ismael Sierra - E_\infty homology of general linear groups and the Goncharov program
  I will present ongoing joint work with A. Kupers and D. Rudenko on the E_\infty-homology of general linear groups and its connection to the Goncharov program and algebraic K-theory. In particular, I will describe a conjectural explicit symbolic formula for the rational algebraic K-theory groups of arbitrary fields. I will also discuss the cases where these conjectures are known to hold, including for number fields, and outline some applications and related open questions.
• Markus Szymik - Beyond general linear groups
  I will outline a framework that extends the context of general linear groups to include symmetric groups, automorphism groups of free groups, the Higman–Thompson groups, and many others. Parts of this talk report on joint work with Anna Marie Bohmann and Nathalie Wahl.
• Emiliano Torti - On a question of Calegari and Venkatesh connecting the (co)homology of modular spaces with algebraic K-theory
  Calegari and Venkatesh constructed, modulo small torsion, a surjection from the degree 2 homology of the rank 2 projective general linear group over a ring of algebraic integers (of odd class number, and with enough embeddings) to the 2nd algebraic K-group of that ring. They asked when this surjection becomes an isomorphism and if it reflects a deeper connection on general degrees. In this talk, we describe a possible way to construct (co)homology classes for such modular spaces starting from large torsion elements in some K-group. This procedure relies on Quillen’s Q-construction and the Steinberg homology groups.
• Raluca Vlad - Tropicalizations of locally symmetric varieties
  Locally symmetric varieties arise as quotients of the form D/G, for D a Hermitian symmetric domain and G a discrete algebraic group. An example of such a variety is the moduli space of abelian varieties of some fixed dimension g. The tropicalization of a locally symmetric variety is a combinatorial object, made out of polyhedral cones, encoding the boundary of a toroidal compactification of the original variety. Using these tropicalizations, we prove results on the (weight 0) cohomology of the moduli of abelian varieties with level structures, and we prove the existence of a Hopf algebra structure on the cohomology of moduli of abelian varieties with complex multiplication. These, in turn, translate into results about the cohomology of general linear groups. Based on joint work with E. Assaf, M. Brandt, J. Bruce, and M. Chan.
• Nathalie Wahl - Stability in the homology of general linear groups, from Quillen to today
  Motivated by the study of algebraic K-theory, Quillen divised 50 years ago a strategy for proving homological stability for groups, applying it to general linear groups. I'll describe Quillen's original approach, and survey some of the developments that have happened since.
• Jennifer Wilson - The Church–Farb–Putman conjecture on the cohomology of the special linear group
  In 2012, Church–Farb–Putman conjectured that the rational cohomology of \(\mathrm{SL}_n(\mathbb{Z})\) vanishes in a range close to its virtual cohomological dimension. I will discuss the current status of this conjecture. This talk will feature work joint with Brück, Miller, Patzt, and Sroka. 

 

Conference Photo

Contact and Organizers

If you have questions about this event, please use the following email address to contact us.

gl.cohomology.muenster@gmail.com

This workshop is organized by

Registration

Registration is now closed.

Support and child care

Childcare is available free of charge for all conference participants.

Accommodation

We kindly ask the participants to arrange their own accommodation. A selection of hotels in Münster can be found here.

SCAM WARNING: Some participants have received scam emails regarding accommodation arrangements. If you are unsure about any emails you received, please contact us.

Venue and Travel Information

Registration for the conference takes place in the ground floor of the Faculty of Mathematics and Computer Science, Einsteinstraße 62 Münster.

University of Münster
Faculty of Mathematics and Computer Science
Einsteinstraße 62
48149 Münster
Germany

The talks take place in room M3 of the building Einsteinstraße 64.

Room SRZ 216/217 on the second floor of the seminar building (Seminarraumzentrum, SRZ) next to the Faculty of Mathematics and Computer Science and the Cluster of Excellence Mathematics Münster is avaiable as a quiet room during the event.

Directions can be found on the University of Münster campus map and on the MM website.

We have also collected practical information in a leaflet: Information for conference guests / Informationsblatt für Tagungsteilnehmer:innen [enIde]

Conference Poster

You are welcome to download the conference poster from this page and display it at your institution.

Sponsors

This conference is supported by the Cluster of Excellence "Mathematics Münster" and the Collaborative Research Centre "Geometry: Deformations and Rigidity".