Abstracts for Conference on "Homology Growth in Topology and Group Theory", May 13 - 17, 2024

We will use Bestvina-Brady Morse theory to analyse maps from cube (or affine) complexes to the circle. The main focus will be on the application on hyperbolic manifolds, and the pursuit of fibrations.
Joint work with Bruno Martelli and Matteo Migliorini.

$L^2$-Betti numbers have played and continue to play an important role in group theory, often as obstructions to some properties of groups. If all $L^2$-Betti  numbers vanish, then one can define a secondary invariant, the $L^2$-torsion, which is the $L^2$-analogue of the classical notion of Reidemeister torsion. It is a more sophisticated and richer but also harder to analyze invariant.  In this talk we want to describe some of its applications and relations to group theory and 3-manifolds which should illustrate its potential.

I will describe several ways of measuring homology growth, and their connections to $L^{2}$-homology and local homology with coefficients in skew fields. I will also discuss some properties of the resulting invariants and some open problems.

Partially based on a joint work with Grigori Avramidi(MPIM) and Kevin Schreve(LSU).

I will report on joint work with Sam Hughes, Caludio Llosa Isenrich, Pierre Py, Ryan Spitler, and Stefano Vidussi that proves, among other things, profinite rigidity of fundamental groups of products of Riemann surfaces among fundamental groups of compact Kähler manifolds. Essential to the proof is profinite rigidity of the collection of homomorphisms to cocompact Fuchsian groups.