Finiteness properties of coabelian subgroups of hyperbolic groups

Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$ if it has a classifying space with finitely many cells of dimension at most $n$, generalising finite generation and finite presentability, which are equivalent to types $F_1$ and $F_2$. Hyperbolic groups are of type $F_n$ for all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties. In recent work with Py, we used methods from Complex Geometry to prove that for every $n>0$ there is a hyperbolic group with a subgroup of type $F_{n-1}$ and not $F_n$. This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. In this talk we will explain this result and present other recent progress on constructing coabelian subgroups of hyperbolic groups with exotic finiteness properties. This talk is based on joint works with Kropholler, Martelli--Py, and Py.