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Nilpotent groups have polynomially bounded homological filling invariants

Gabriel Pallier

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.

Nilpotent groups have polynomially bounded homological filling 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.

Paper Structure

This paper contains 4 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Proof of Theorem \ref{['prop:high-nil-fill']}
  4. Final remarks

Key Result

Theorem 1.1

Let $\Gamma$ be a finitely generated nilpotent group of cohomological dimension $n$ over $\mathbf Q$. Then the Dehn function and the homological filling functions $\operatorname{cFV}^d_\Gamma$ with integer coefficients are polynomially bounded for all $d \in \{2, \ldots, n\}$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1: Polynomial similarity
  • proof
  • Lemma 2.2: Polynomial distortion
  • proof
  • Lemma 2.3: Euclidean dilation
  • proof