Bounding the homological finiteness length

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 n-dimensional 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.