# Talks winter 2024/25

### 10 Oct - Sam Shepherd

tba

### 17 Oct - Silke Meißner

tba

### 31 Oct - Tingxiang Zou

tba

### 7 Nov - Sam Hughes

tba

Thursday 11-12, room tba

### 10 Oct - Sam Shepherd

tba

### 17 Oct - Silke Meißner

tba

### 31 Oct - Tingxiang Zou

tba

### 7 Nov - Sam Hughes

tba

Thursday 11-12, SR 4

### 11 Apr - Kevin Iván Piterman. A categorical approach to study posets of decompositions into subobjects

Given a sequence of groups \(G_n\) with inclusions \(G_n \to G_{n+1}\), an important question in group (co)homology is whether there is homological stability. That is, if for a given integer j, there is some m such that for all n>m, the map \(H_j(G_n) \to H_j(G_{n+1})\) is an isomorphism. To detect this behaviour one usually constructs a family of highly connected simplicial complexes \(K_n\) on which the groups \(G_n\) naturally act. For example, for the linear groups \(GL_n\) or \(SL_n\), \(K_n\) can be the Tits building or the complex of unimodular sequences, while for the automorphism group of the free groups \(F_n\) one can take the complex of free factors.

In this talk, we discuss a categorical framework that describes these constructions in a unified way. More precisely, for an initial symmetric monoidal category C, we take an object X and consider the poset of subobjects of X. From this bounded poset, we take only those subobjects which are complemented, i.e. \(x \vee y = 1\) and \(x \wedge y = 0\), and the join operation coincides with the monoidal product. The monoidal product should be interpreted as the "expected" coproduct of the category. Thus, for the free product in the category of groups, if we start with a free group of finite rank then the complemented subobject poset is exactly the poset of free factors, and for the category of vector spaces with the direct sum we obtain the subspace poset. From this construction, we define related combinatorial structures, such as the poset of (partial) decompositions or the complex of partial bases, and establish general properties and connections among these posets. Finally, we specialise these constructions to matroids, modules over rings, and vector spaces with non-degenerate forms, where there are still many open questions.### 18 Apr - Thomas Koberda. Using logic to study homeomorphism groups

I will describe some recent results on the first order rigidity of homeomorphism groups of compact manifolds, and their applications to dynamics of group actions on manifolds. I will also describe how to find "syntactic" invariants of manifolds, and how these can be used to give a conjectural model-theoretic characterization of the genus of a surface. I will explain some of the details of the proof, including how homeomorphism groups of manifolds interpret second order arithmetic in a uniform manner.

### 25 Apr - Jerónimo García-Mejía. Dehn functions of nilpotent groups

Since Gromov's celebrated polynomial growth theorem, the understanding of nilpotent groups has become a cornerstone of geometric group theory. An interesting aspect is the conjectural quasiisometry classification of nilpotent groups. One important quasiisometry invariant that plays a significant role in the pursuit of classifying these groups is the Dehn function, which quantifies the solvability of the world problem of a finitely presented group. Notably, Gersten, Holt, and Riley's work established that the Dehn function of a nilpotent group of class \(c\) is bounded above by \(n^{c+1}\).

In this talk, I will explain recent results that allow us to compute Dehn functions for extensive families of nilpotent groups arising as central products. Consequently, we obtain a large collection of pairs of nilpotent groups with bilipschitz equivalent asymptotic cones but with different Dehn functions.

This talk is based on joint work with Claudio Llosa Isenrich and Gabriel Pallier.

### 2 May - Pierre Touchard. On Transfer Principles for Mekler Groups

This talk is about a joint work with Aris Papadopoulos and Blaise Boissonneau [BPT].

Starting from any first order structure S, Mekler constructs in [M] a 2-nilpotent group of prime exponent M=(G, ·) which interprets, in the pure language of groups, the structure S. This 2-nilpotent group shares numerous model theoretical properties with the structure S, notably in terms of dividing lines:

M is Stable (resp. Simple, NIP_n for every n, Strong, NTP2...) if and only if S satisfies this property. See [CH].

I will motivate these results and show how one can generalise some of them, by considering a uniform hierarchy of dividing lines, introduced in [GHS]: the NC_K-hierarchy, which rises from coding (or not coding) Ramsey classes of structures K . I will also state a transfer principle for stably embedded pairs of Mekler groups (all these notions will be defined). Our method, that I will briefly sketch, was to establish new relative quantifier elimination results, and was inspired by a step-by-step approach for proving transfer principles in valued fields.

[BPT] Boissonneau, Papadopoulos and T., Mekler's Construction and Murphy's Law for 2-Nilpotent Groups, arXiv:2403.20270.

[GHS] Guingona, Hill and Scow, Characterizing model-theoretic dividing lines via collapse of generalized indiscernibles.

[M] Mekler, Stability of nilpotent groups of class 2 and prime exponent.

[CH] Chernikov and Hempel, Mekler's construction and generalized stability.

### 6 June - Margarete Ketelsen. Model-theoretic tilting

The tilting construction (as introduced by Fontaine) provides a way to transfer theorems between the worlds of characteristic zero and positive characteristic. Classically, this was done for perfectoid fields: for each perfectoid field of characteristic zero, we can obtain its tilt - a perfectoid field of positive characteristic. Perfectoid fields are complete non-discretely valued fields of rank 1 that satisfy some perfectness condition. In my talk, I will tell you how we can extend the tilting construction to certain valued fields of higher rank using model theory. The model-theoretic tilt we obtain is only defined up to elementary equivalence, so we tilt the theories rather than the fields themselves.

To understand a valuation of higher rank, we can decompose it into simpler parts. I will introduce the standard decomposition and tell you about ongoing joint work with Sylvy Anscombe and Franziska Jahnke, where we prove for certain valued fields that the theories of the fields in the standard decomposition only depend on the theory of the valued field we started with. This work is crucial for the well-definedness of the model-theoretic tilt.

### 13 June - Francesco Fournier-Facio. Bounded cohomology of transformation groups of \(\mathbb{R}^n\)

Bounded cohomology is a functional analytic analogue of group cohomology, with many applications in rigidity theory, geometric group theory, and geometric topology. A major drawback is the lack of excision, and because of this some basic computations are currently out of reach; in particular the bounded cohomology of some “small” groups, such as the free group, is still mysterious. On the other hand, in the past few years full computations have been carried out for some “big” groups, most notably transformation groups of \(\mathbb{R}^n\), where the ordinary cohomology is not yet completely understood. I will report on this recent progress, which will include joint work with Caterina Campagnolo, Yash Lodha and Marco Moraschini, and joint work with Nicolas Monod, Sam Nariman and Alexander Kupers.

### 20 June - Konstantin Recke. Percolation on groups

Originating in statistical physics as a model of a porous medium, Bernoulli percolation has become a fundamental model in probability theory. In classical Bernoulli percolation, edges (or vertices) of \(\mathbb Z^d\) are deleted independently of each other and with fixed survival probability \(p\in[0,1]\). Despite significant progress in understanding this model, basic questions remain, constituting some of the most perplexing problems in probability theory.

In the late 1990s, Benjamini, Lyons, Peres, and Schramm initiated a program to study Bernoulli percolation and other invariant percolation models (i.e., random subgraphs whose distribution is invariant under some natural group action) on graphs beyond \(\mathbb Z^d\). Cayley graphs of infinite groups provide a rich class of examples, and the behavior of percolations turns out to be closely related to the geometric properties of the underlying group

This talk will start with a brief introduction to Bernoulli percolation, highlighting what is known as well as open questions. I will then give a gentle introduction to the aforementioned program, focussing on its main questions and the different motivations behind. I will provide a glimpse into some of the fascinating mathematics involved, primarily by reviewing the case of amenable groups. Finally, I will present recent progress beyond amenability based on joint work with Chiranjib Mukherjee.

### 27 June - Mathias Stout. Tameness for ordered fields with real analytic structure

It is well-known that the subanalytic structure on the real numbers is o-minimal. Cluckers and Lipshitz have shown that this remains true for elementary extensions of the reals equipped with certain nonstandard analytic functions. More precisely, if B is a real Weierstrass system, then any real closed field with B-analytic structure is o-minimal.

In this talk, we consider B-analytic structure on ordered fields that are not necessarily real closed. Such structures cannot be o-minimal in general. Still, they turn out to be tame as valued fields: when equipped with a convex valuation, they give rise to new examples of ω-h-minimal structures.

This talk is based on joint work with Kien Huu Nguyen and Floris Vermeulen.### 4 July - Davide Carolillo. On the homogeneity of uncountable relatively free groups in varieties

Recently Sklinos proved that every uncountable free group is not \(\aleph_1\)-homogenenous. This was later generalised by Belegradek to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary.

In this talk, we will present a joint work with G. Paolini that answers Belegradek's question in the negative by using some techniques inspired by the classical analysis of relatively free groups in infinitary logic. Our methods are general: they also apply to varieties with torsion and, more generally, to any variety of algebras in a countable language. For instance, we can prove that if V is a group variety containing a non-solvable group, then any uncountable V-free group is not \(\aleph_1\)-homogenenous, and if R is a countable ring, then any uncountable free left R-module is \(\aleph_1\)-homogeneous if and only if R is left perfect.

After discussing some of the main applications of our results to the homogeneity problem, we will show how the same techniques can be used to study the relationship between two notions of strong substructure in free algebras: the syntactic notion of elementary substructure and the algebraic notion of V-free factor.Finally, we will give a structure theorem characterising the homogeneity of uncountable free algebras in a countable language in terms of an easily verifiable combinatorial condition. This theorem gives a new perspective on some famous results of Eklof, Mekler and Shelah on the topic of almost free algebras, showing that the problem of establishing the (first-order) homogeneity of uncountable free algebras corresponds exactly to the problem of determining the axiomatizability of the class of free algebras and the number of non-isomorphic models of their theory in some suitable infinitary logics.

Thursday 11-12, SRZ 216/17 (if not announced differently)

### 12 Oct - Rosario Mennuni. Some definable types in the wild

While definable types are usually studied in "tame" contexts, their usefulness and amenability to model-theoretic investigation even "in the wild" is, historically, not a surprise: for instance, Lascar defined the tensor product of definable types by generalising the existing notion on ultrafilters, which may be viewed as (trivially) definable types in the richest possible language on a given set.

By pushing the tensor product forward along the addition, one shows that the usual sum of integers may be extended to the space of ultrafilters over Z, yielding a compact right topological semigroup. The analogous construction also goes through for the product, and these facts had important applications in additive combinatorics and Ramsey theory.

Recently, B. Šobot introduced two (ternary) notions of congruence on the space above. I will talk about joint work with M. Di Nasso, L. Luperi Baglini, M. Pierobon and M. Ragosta, in which the study of these congruences led us to isolate a class of ultrafilters enjoying characterisations in terms of tensor products, directed sets, profinite groups, and more.### 19 Oct - Alessandro Codenotti. Ranks for tame dynamical systems and model theory

By slightly generalizing a notion of rank originally introduced in the context of metric tame dynamical systems by Glasner and Megrelishvili, we define an ordinal valued rank for the action of Aut(M) on the space of types over M, where M is a model of some theory T. We then investigate the relationship between this rank and the dividing lines in the model theoretic hierarchy, in particular we characterize NIP theories as those with non-infinite rank and stable theories as those with rank 0. This is joint work with Daniel Max Hoffmann.

### 2 Nov - Benjamin Brück. Top-degree cohomology in the symplectic group of a number ring

I will indicate how one can use Tits buildings to show that the "top-degree" cohomology of Sp2n(R), the symplectic group over a number ring R, depends on number theoretic properties of R:

In recent work with Himes, we proved that Sp2n(R) has non-trivial rational cohomology in its virtual cohomological dimension if R is not a principal ideal domain. We gave a lower bound for the dimension of these cohomology groups in terms of the class number of R. This contrasts results joint with Santos-Rego-Sroka that show that the top-degree cohomology group of Sp2n(R) is trivial if R is Euclidean.

Both of these results have counterparts in the setting of SLn(R) that were established by Church-Farb-Putman. A key ingredient in all of this is the action of these groups on associated Tits buildings.### 9 Nov - Marco Amelio. Non-split sharply 2-transitive groups in odd characteristic

Until recently, the existence of non-split sharply 2-transitive groups (i.e., sharply 2-transitive groups without a normal abelian subgroup) was an open problem. The first examples of such groups were exhibited by Rips, Segev and Tent in 2017 and by Rips and Tent in 2019. It is possible to associate to every sharply 2-transitive group a characteristic that is either 0 or a positive prime number. The first of these examples were in characteristic 2, while the others were in characteristic 0, leaving the problem open for odd characteristics. In this talk, I will outline recent progress made in adapting the construction in characteristic 0 to build examples of non-split sharply 2-transitive groups in odd characteristic using methods of geometric small cancellation. I will also give a rough explanation of how these methods relate to the usual small cancellation conditions for group presentations. This is joint work with Simon André and Katrin Tent.

### 23 Nov - Yuri Santos Rego. Reflection groups and their finite quotients

There has been growing interest in the following question: to what extent do finite quotients of a given group determine its structure? For instance, do profinite completions detect specific properties, isomorphism types, or elementary theories?

In this talk we shall review some concepts and known results and problems around the above mentioned question, and then shift focus to the state of knowledge for the family of Coxeter groups. Based on joint work with Petra Schwer.### 30 Nov - Simone Ramello. Definable henselian valuations in positive residue characteristic

**Takes place in 120.029 / 120.030**(second floor Orleansring 10)!Jahnke and Koenigsmann started a classification of when a henselian valuation is definable on a henselian field, providing a full characterization for the case where the canonical henselian valuation has residue characteristic zero. We extend their work to the case where the residue characteristic is positive, drawing on the toolkit of independent defect to summon a definable henselian valuation out of certain defect extensions. This is joint work with Margarete Ketelsen (Münster) and Piotr Szewczyk (Dresden).

### 7 Dec - Katrin Tent. On the model theory of free and open generalized polygons

We show that for any \(n\geq 3\) the theory of free generalized \(n\)-gons is complete and strictly stable yielding a new class of examples in the zoo of stable theories exhibiting many of the properties of free groups with very elementary proofs.

Joint work with A.-M. Ammer.

### 14 Dec - Martin Bays. A group action version of the Elekes-Szabó Theorem

I will show how to strengthen the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, by weakening the combinatorial hypothesis to an asymmetric situation modelled on a group action; the proof will go via first obtaining then disposing of an algebraic group action. This is recent work with Tingxiang Zou.

### 18 Jan - Anna de Mase. Value groups of finitely ramified henselian valued fields and model completeness

A result obtained by J. Derakhshan and A. Macintyre states that the theory of a mixed characteristic henselian valued field with finite ramification, and whose value group is a Z-group, is model complete in the language of rings if the theory of the residue field is model complete in the language of rings. In this talk, we will see how this result can be generalized to mixed characteristic henselian valued fields with finite ramification, but with different value groups. We will address the case in which the value group is an ordered abelian group with finite spines, and (if time permits) the case in which it is elementarily equivalent to the lexicographic sum of Z with a minimal positive element. In both cases, we give a one-sorted language (expansion of the language of rings) in which the theory of the valued field is model complete if the theory of the residue field is model complete in the language of rings.

### 25 Jan - Zahra Mohammadi. Model theoretic aspects of the free factor complex

Free Factor Complexes are important objects in Geometric Group Theory. We are studying the model theoretic aspects of these complexes. Bestvina and Bridson showed that the automorphism group of the free factor complex is naturally isomorphic to the automorphism group of the free group, respectively Out(Fn) for the complex of conjugacy classes of free factors. We obtained a model theoretic proof for this result and proved that the free factor complexes are homogeneous in the sense of model theory. Our next aim is to construct saturated models of these complexes in order to study them from the point of view of stability.