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.

Revised version

References in corpus (3)

• Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem
• Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices
• Infinite generation of non-cocompact lattices on right-angled buildings

Cited by in corpus (7)

• Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem
• Graphs and complexes of lattices
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• On the finiteness length of some soluble linear groups
• Finite presentability of Kac-Moody groups over finite fields
• Semidualities from products of trees
• A note on the rational homological dimension of lattices in positive characteristic