Geometric Group Theory Seminar

Elia Fioravanti (Karlsruher Institut für Technologie) 22.10.25

Automorphisms of virtually special groups
For which families of finitely generated groups G are automorphisms "tame"? For instance, one can wonder whether Out(G) is finitely generated, or whether the stretch factors of the automorphisms of G are algebraic integers. Following breakthroughs of Rips and Sela in the 90s, automorphisms of hyperbolic groups are known to satisfy both of these forms of tameness, but much less is known on automorphisms of more general non-positively curved groups. We show that automorphisms of compact special groups (in the sense of Haglund and Wise) also satisfy the same two forms of tameness. Proofs rely on non-small R-trees and a new JSJ decomposition over centralisers.

Andrew Ng (Universität Bonn) 29.10.25

Fibring hyperbolic Coxeter groups with finitely presented kernel
Following Agol’s work on virtual fibring of 3-manifolds, there has been substantial interest in both fibring higher dimensional manifolds and algebraic fibrings of finitely generated groups. Algebraic fibrings of word-hyperbolic groups have also lead to the discovery of subgroups with exotic finiteness properties. These two lines of research met in the work of Italiano-Martelli-Migliorini in the construction of a hyperbolic group with a non-hyperbolic subgroup of type F as well as a fibred 5-dimensional hyperbolic manifold, which is based on the theory of hyperbolic Coxeter groups. Previous work has produced many examples of fibred hyperbolic Coxeter groups, but finitely presentable examples have been relatively scarce. In joint work with Italiano and Migliorini, we construct infinitely many quasi-isometry classes of hyperbolic Coxeter groups in every vcd at least 3 that fibre with finitely presented kernel and obtain many examples of subgroups of type \(F_2\) but not \(F_3\).

Motiejus Valiunas (Uniwersytet Wrocławski) 5.11.25

Means on groups and degrees of commutativity
Given a finite group \(G\), one can count the proportion of pairs of elements in \(G\) that commute -- giving a number, denoted \(\mathrm{dc}(G)\), behaviour of which has been studied since the 1960s. Such a notion has straightforward generalisations to residually finite or amenable groups. To make sense of this for other infinite groups G, one needs to define some sort of a nice measure or a mean on \(G\). In this talk, I will explain how such means -- more specifically, finitely additive probability means that give the "correct" answer for cosets of subgroups -- can be constructed. As an application of these methods, one can define \(\mathrm{dc}(G)\) for any group G, and show that \(\mathrm{dc}(G) > 0\) if and only if \(G\) is finite-by-abelian-by-finite. This is joint work with Armando Martino.

Jean Raimbault (Aix Marseille Université) 12.11.25

Arithmetic link complements
Two classical constructions of cusped hyperbolic 3--manifolds of finite volume are (some) link complements in the sphere, and quotients of hyperbolic space by congruence subgroups of Bianchi groups such as \(\mathrm{PSL}_2(\mathbb{Z}[i])\). A conjecture of Baker and Reid posits that only finitely many manifolds occur as both. I will discuss this conjecture in relation with well-known and conjectured properties of arithmetic groups, and present a proof of the conjecture obtained in joint work with S. Kionke.

Waltraud Lederle (Universität Bielefeld) 19.11.25

Invariant random subgroups, elementwise conservative actions and boomerang subgroups
Invariant random subgroups, which are probability measures on sets of subgroups, have been an important subject of research in the last 15 years and can be thought of as point stabilizers of probability measure preserving (p.m.p.) actions. We are introducing elementwise conservative (EC) actions as generalizations of p.m.p. actions, and boomerang subgroups as "generic instances" of EC actions. I will talk about what these objects are, what they have in common and what tells them apart. Topics that we will adress in this talks are Margulis' normal subgroup theorem and the Stuck-Zimmer rigidity thorem, Poincaré recurrence, non-singular dynamics, Schreier graphs and critical exponents or growth rates. No prior knowledge on the subject is required. Based on joint works with Yair Glasner and Tobias Hartnick

Ian Leary (University of Southampton) 26.11.25

Comparing commutator subgroups of right-angled Artin and Coxeter groups
(Joint work with Nansen Petrosyan.) Finite-index subgroups of Coxeter groups and co-abelian subgroups of right-angled Artin groups provide a wealth of examples in group cohomology and geometric topology. I shall explain the close connection between these two families of groups.

No talk (dies academicus), 3.12.25

Naomi Andrew (Université Paris Saclay) 10.12.25

The Farrell–Tate K-theory of Out\((F_n)\)
Given a (nice enough) group, there is an isomorphism, due to Lück, relating the rationalised \(K\)-theory groups of its classifying space to a large product of cohomology groups, some with rational and some with \(p\)-adic coeffecients. In recent joint work with Irakli Patchkoria, we identify a generalised cohomology theory capturing the \(p\)-adic part of this product. Working in Out\((F_n)\), in ranks close to \(p\) we can fully compute this \(p\)-adic part and in this way produce an infinite family of odd-dimensional summands in the rationalised \(K\)-theory of Out\((F_n)\). I will discuss these results and the tools that go into them, which range from spherical group rings to the lemma that is not Burnside's, via results about centralisers in Out\((F_n)\): I will try to explain how all these various ideas fit together!

Alessandro Sisto (Oilthigh Heriot-Watt), 17.12.25

Towards geometric finiteness in mapping class groups
Given a subgroup of a mapping class group, there is a corresponding extension group, and fundamental groups of surface bundles (with injective monodromy) are exactly these extension groups. Farb and Mosher introduced the notion of convex-cocompact subgroup of a mapping class group, and a subgroup is convex-cocompact if and only if the corresponding extension group is hyperbolic. Moving beyond this, one may wonder what happens for subgroups that are "close" to convex-cocompact. I will discuss some cases where the corresponding extension groups are hierarchically hyperbolic; I will explain what this means, what consequences this has, and speculate about notions of geometric finiteness in mapping class groups. Based on joint works with Spencer Dowdall, Matt Durham, and Chris Leininger, and Jacob Russell.

Xiyan Zhong (MPI Bonn) 7.1.26

Prym Representations and Twisted Cohomology of the Mapping Class Group with Level Structures
The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-\(\ell\) mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed. We prove this infinitesimal rigidity by calculating the twisted cohomology of the level-\(\ell\) mapping class group with coefficients in the Prym representation, and more generally in the \(r\)-tensor powers of the Prym representation. Our results also show that when \(r\geq 2\), this twisted cohomology does not satisfy cohomological stability, i.e. it depends on the genus \(g\).

Piotr Mirzeka (Uniwersytet im. Adama Mickiewicza w Poznaniu) 14.1.26

Sampling elements of a finite group: efficiency of the product replacement algorithm with accumulator
Let \(G\) be a finite group generated by \(k\) elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of \(G\). We study a refinement of this algorithm that is designed to output individual elements of \(G\). We show that after \(O(k^2\log|G|)\) steps, the distribution of the output is close to uniform on~\(G\), which improve upon the best results known to date. The proof proceeds via spectral gap estimates and uses computer assisted calculations. This is a joint work with Michał Marcinkowski.

Giorgio Mangioni (Oilthigh Heriot-Watt) 21.1.26

Dehn fillings of short HHG, and residual properties of Artin groups
The Dehn Filling theorem of Osin and Groves-Manning is a powerful tool to produce a plethora of (relatively) hyperbolic quotients of relatively hyperbolic groups. We present an analogue of this procedure for the class of short HHG, which include several Artin groups, graph manifold groups, and more. Rather than getting into the details of the result (or even defining what a short HHG is), we show how these tools can be applied to prove that "generic" Artin groups, in a suitable probabilistic sense, are residually hyperbolic and Hopfian. This sheds more light on the famous conjecture of residual finiteness of Artin groups. This talk is based on joint work with Alex Sisto.

Holger Kammeyer (Heinrich Heine University Düsseldorf) 28.1.26

Profiniteness of higher rank volume
While many properties such as amenability, property (T) and FA, finiteness properties, Euler characteristic, ... can differ for profinitely isomorphic groups, we present a result in the positive direction: The covolume of lattices in higher rank Lie groups with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a profinite invariant of octonionic hyperbolic congruence manifolds. Joint work with Steffen Kionke and Ralf Köhl.

Cornelia Drutu (University of Oxford/MPI Bonn) 4.2.26

Isoperimetric inequalities in CAT(0) spaces
In this talk, I shall explain the existence, in non-positively curved spaces, of a 2-dimensional isoperimetric gap, similar to the one in dimension 1, i.e. for the filling of loops by discs. The argument is based on the thinness of a particular type of tetrahedra, the minimal tetrahedra. This is joint work with Urs Lang, Panos Papasoglu and Stephan Stadler.

Oberseminar Geometric Group Theory

Winter semester 2025/26