Alternatively have a look at the program.

## Groups of type FP via graphical small cancellation II: relators with symmetry

Tom Brown and I gave a construction of non-finitely presented groups of type FP that is independent of the Bestvina-Brady argument but instead used graphical small cancellation. Our result required the construction of a 'Spectacular complex' - a 2-dimensional acyclic complex whose polygons have some small cancellation property. A new argument produces more examples including examples of larger cohomological dimension.

(Builds on joint work with Tom Brown)

## Algebraic fibering and incoherence of Coxeter groups

I will discuss applications of Bestvina-Brady Morse theory to construction of virtual algebraic fiberings and incoherence of groups, with focus on Coxeter groups. This will include joint work with Norin and Wise.

## Fibring, Algebraic Fibring, and Hyperbolic Manifolds

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.

## Galois cohomology and the sign of the Euler characteristic

$L^2$-Betti numbers capture free homology growth of a group along certain sequences of finite index normal subgroups.

## Survey on $L^2$-torsion and its (future) applications

$L^2$-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.

## Homology torsion growth in polynomially-growing mapping tori

We show that mapping tori of polynomially-growing automorphisms of many non-positively curved groups have vanishing homology torsion growth. In the case of free-by-cyclic groups with polynomially-growing monodromy, this confirms a conjecture of Lück relating the integral torsion and the $L^2$-torsion of a group. Our main tool is the cheap rebuilding property introduced by Abert—Bergeron—Fraczyk—Gaboriau. This is based on joint work with Naomi Andrew, Yassine Guerch and Sam Hughes.

## Homology growth: an overview

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).

## All triangulations have a common stellar subdivision

We address two longstanding open problems, one originating in PL topology, another in birational geometry. First, we prove the weighted version of Oda’s strong factorization conjecture (1978), and prove that every two birational toric varieties are related by a common iterated blowup (at rationally smooth points). Second, we prove that every two PL homeomorphic polyhedra have a common stellar subdivisions, as conjectured by Alexander in 1930.

## Profinite rigidity amongst Kähler groups

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.

## A new instance of the Lück approximation in the positive characteristic

Let $\Gamma$ be a group and let $K$ be a field. For every matrix $A\in \text{Mat}_{n\times m}(K[\Gamma])$ and every normal subgroup $N$ of $\Gamma$ of finite index let us define $$\begin{array}{cccc} \phi_{\Gamma/N}^A: & K[\Gamma/N]^n & \to & K[\Gamma/N]^m \\ &(x_1,\ldots, x_n)&\mapsto & (x_1,\ldots, x_n)A\end{array}.$$ This is a $K$-linear map between two finite-dimensional $K$-vector spaces. Thus, we can define a Sylvester rank function of $K[\Gamma]$ by means of $$\text{rk}_{\Gamma/N}(A)=\frac{\text{dim}_K \text{Im} \phi_{\Gamma/N}^A}{|\Gamma:N|}.$$

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