This summer school on homological methods and their applications to groups and topology aims to expose young researchers to homological techniques used in geometric and topological group theory nowadays.
Three lecturers will each give an in-depth introduction to an aspect of group cohomology during a four-hour mini-course (with one exercise session).
The summer school is organized through the Ghent University Doctoral school training program.

Dawid Kielak: Fibring over the circle and L²-homology

I will introduce L²-homology and Novikov homology, show how they interact in the setting of RFRS groups, and finally how it all combines to give us a homological way of detecting when RFRS groups virtually algebraically fibre. I will discuss how this can be used to understand virtual fibring over the circle for manifolds of low dimension.

Steffen Kionke: An Introduction to arithmetic groups and their cohomology

Arithmetic groups are (roughly speaking) groups of integral points of linear algebraic groups. They are well-studied and useful examples in group theory, they are central players in number theory and in the theory of automorphic forms and they are interesting from a geometric point of view based on their properly discontinuous actions on symmetric spaces.
This mini-course provides a short introduction to arithmetic groups and the associated locally symmetric spaces. We will discuss homological properties of arithmetic groups. In particular, we will explain how to calculate the Euler characteristic and L²-Betti numbers of arithmetic groups building on Harder‘s Gauß-Bonnet formula. The mini-course will conclude with an outlook to cohomology growth of groups in p-adic analytic towers.

Conchita Martinez-Perez: Homological aspects of right-angled Artin groups

Right-angled Artin groups are a fascinating class of groups well known for many reasons. These groups, thanks to the Bestvina-Brady construction provided counter-examples for important open questions about homological finiteness conditions in groups, and they enjoy very useful homological properties that very often can be described nicely in terms of the combinatorics of the defining graph. In this mini-course we will describe some of this properties and consider also some possible generalizations to other related families, such as families of groups of automorphisms or Artin groups.

This schedule is still subject to changes.

The list of registered participants can be found here.

• The summer school takes place from Tuesday, June 10 until Friday, June 13, 2025.
• All lectures take place at Ghent University, Campus Sterre, building S1, Lecture room 3.3.

With the support of the Flemish Government.