Bounding the homological finiteness length

TL;DR

This paper introduces a new criterion for bounding the homological finiteness length of HF-groups, with applications to lattices, arithmetic groups, and simple groups over number fields, solving several existing conjectures.

Contribution

It provides a novel criterion for bounding homological finiteness length and applies it to various group classes, resolving multiple open conjectures.

Findings

Bound on homological finiteness length for non-uniform lattices in n-dimensional complexes

Upper bounds for arithmetic groups over function fields

Verification of a conjecture for simple groups over number fields

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.