Subgroups of hyperbolic groups and finiteness properties: a complex hyperbolic approach

In this thesis we give positive answer to an old question by Noel Brady about some finiteness properties of subgroups of hyperbolic groups. More precisely, following a recent article by Claudio Llosa Isenrich and Pierre Py, we show that, for every n>1, there exists an infinite family (G_{n,j}) of pairwise non-commensurable hyperbolic groups and surjective homomorphisms f_{n,j}:G_{n,j}->Z such that, for every j, the subgroup Ker f_{n,j} of G_{n,j} satisfies the finiteness property of type F_{n-1} but not of type F_n. In order to achieve the desired result, we use techniques from complex geometry and the theory of arithmetic lattices and mix them with known results about the homotopical BNSR invariants of a group G, which were introduced by Burkhardt and Renz. Inside the group PU(n,1) of biholomorphic isometries of the n-dimensional complex hyperbolic space, we will exhibit an infinite family of pairwise non-commensurable hyperbolic groups, the so-called arithmetic lattices of the simplest type, which satisfy the aforementioned property.