Nilpotent groups have polynomially bounded homological filling invariants

TL;DR

This paper proves that all finitely generated nilpotent groups have polynomially bounded homological filling invariants, confirming Gromov's claim and providing explicit polynomial bounds across all degrees.

Contribution

It offers a unified proof, based on Gromov's hints, establishing non-optimal polynomial bounds for homological filling invariants in all degrees for all finitely generated nilpotent groups.

Findings

Polynomial bounds are established for all degrees.

The proof applies to all finitely generated nilpotent groups.

It confirms Gromov's claim about polynomially bounded invariants.

Abstract

Gromov claimed, with a sketch of proof, that simply connected nilpotent Lie groups have polynomially bounded filling invariants. The literature establishes this, often with a stronger conclusion where the exponent of polynomiality is computed or estimated, for some classes of nilpotent groups, or ranges of filling degrees. We provide a proof, in part based on Gromov's hints, yielding at once (non-optimal) polynomial upper bounds on the homological filling invariants in every degree for all finitely generated nilpotent groups, or equivalently, for all simply connected nilpotent Lie groups having lattices.