Topological invariants of groups and Koszul modules

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