Topological invariants of groups and Koszul modules

TL;DR

This paper establishes algebraic bounds on Koszul modules and applies these results to derive new bounds on various group invariants, such as Alexander invariants and Chen ranks, for groups related to topology and geometry.

Contribution

It provides a uniform vanishing theorem and sharp Hilbert function bounds for Koszul modules, with applications to group invariants in topology and algebraic geometry.

Findings

Bounded Alexander invariants in terms of first Betti number

Derived upper bounds for Chen ranks

Established vanishing results for Koszul modules

Abstract

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves; (2) nilpotent fundamental groups of compact Kaehler manifolds; (3) the Torelli group of a free group.