Finiteness Properties of Non-Uniform Lattices on CAT(0) Polyhedral Complexes

TL;DR

This paper establishes an upper bound on the homological finiteness length of non-uniform lattices on CAT(0) polyhedral complexes, providing new insights into their algebraic and geometric properties.

Contribution

It introduces a new bound for the finiteness length of non-uniform lattices on CAT(0) complexes, simplifying previous proofs and advancing understanding of arithmetic groups over function fields.

Findings

Homological finiteness length of non-uniform lattices is less than the complex's dimension.

Provides an upper bound for arithmetic groups over function fields.

Simplifies proof of a longstanding conjecture by Bux and Wortman.

Abstract

We show that the homological finiteness length of a non-uniform lattice on a locally finite CAT(0) n-dimensional polyhedral complex is less than n. As a 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.