Abstract. We give a criterion for bounding the homological finiteness length of certain HF-groups. This is used in two distinct contexts. Firstly, the homological finiteness length of a non-uniform lattice on a locally finite ndimensional contractible CW-complex is less than n. In dimension two it solves a conjecture of Farb, Hruska and Thomas. As another corollary, we obtain an upper bound for the homological finiteness length of arithmetic groups over function fields. This gives an easier proof of a result of Bux and Wortman that solved a long-standing conjecture. Secondly, the criterion is applied to integer polynomial points of simple groups over number fields, obtaining bounds established in earlier works of Bux, Mohammadi and Wortman, as well as new bounds. Moreover, this verifes a conjecture of Mohammadi and Wortman.

We show that the Farrell-Jones Conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag-Solitar groups.the Deutsche Forschungsgemeinschaft (DFG) through the Berlin Mathematical School (BMS). The

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.Comment: final versio

We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the Fcohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every finitely generated regular branch group has infinite rational cohomological dimension. Moreover, we prove that the first Grigorchuk group G is not contained in Kropholler's class HF.