# Cohomological methods in Group Theory

Friday, 23rd September 2022, 1pm – 5pm, Senate House, room 102 (Senate House University of London, Malet St, London WC1E 7HU). Local organiser: Brita Nucinkis

The meeting will take place in person at University of London (Central London Campus).

## Speakers

• Paula Macedo Lins de Araujo (University of Lincoln), Soluble matrix groups and $R_\infty$ property
• Yuri Santos Rego (University of Magdeburg), Cohomological finiteness length of matrix groups
• Pavel Zalesski (University of Brasilia), Splitting of pro-p groups (as an amalgam or HNN)

To register for the event and to receive the talk links, please email Brita Nucinkis (Brita.Nucinkis@rhul.ac.uk).

The FCG Research Group is supported by an LMS Joint Research Groups in the UK Scheme 3 grant. Limited funding is available for PhD students, allocated on a first come first served basis.

For UK-based mathematicians with caring duties the LMS has a Caring Supplementary Grant scheme which allows participants of meetings like ours to apply for help covering caring costs.

## Abstracts

Paula Macedo Lins de Araujo (University of Lincoln), Soluble matrix groups and $R_\infty$ property

A group automorphism $\varphi: \Gamma \to \Gamma$ induces the action $g\cdot x= gx\varphi(g)^{-1}$ on $\Gamma$. The orbits of such action are called Reidemeister classes.

In the past few years, the sizes of such classes have been intensively investigated. One of the main goals in the area is to classify groups where all classes are infinite. Such groups are said to satisfy the $R_\infty$ property.

Although much is known about Reidemeister classes of nilpotent or polycyclic groups, Reidemeister classes of soluble groups in general have not been as intensively studied, so that there are still has many open questions to be explored.

In this talk, we will discuss the relation of Reidemeister classes with non-abelian cohomology. We also show conditions for certain soluble groups to have the $R_\infty$

This talk is based on joint works with K. Dekimpe and Y. Santos Rego.

Yuri Santos Rego (University of Magdeburg), Cohomological finiteness length of matrix groups

Since many groups occurring naturally turn out to be linear (e.g., surface groups, Coxeter groups, crystallographic groups), the family of groups defined by matrices has rich (co)homological properties. The cohomological finiteness length is a particularly useful quasi-isometry invariant, though computing it for prominent matrix groups is an ongoing challenge.

In this talk we shall recall the definition and important properties of the cohomological finiteness length, going through well-known examples. Shifting focus, we shall discuss the state of knowledge for classical matrix groups over integral domains and present some recent results in this direction.

Pavel Zalesski (University of Brasilia), Splitting of pro-p groups (as an amalgam or HNN)

I shall talk about Stallings type results of splittings of pro-p groups as an amalgamated free pro-p product or a pro-p HNN-extension as well as some other aspects of such splittings.