  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.
