# Talks summer 2024

### 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.

### 13 June - Francesco Fournier-Facio. Title tba

Abstract: