Torsion homology growth of polynomially growing free-by-cyclic groups

TL;DR

The paper establishes that for free-by-cyclic groups $\Gamma = F_m \rtimes_\varphi \mathbb{Z}$ with polynomially growing monodromy, the homology torsion growth vanishes in all dimensions for any Farber sequence, hence $\rho^\mathbb{Z}(\Gamma) = \rho^{(2)}(\Gamma) = 0$ and validating a Lück conjecture in this setting. The approach hinges on the cheap $\alpha$-rebuilding property and a graph-of-groups decomposition derived from a splitting result for polynomially growing monodromy, enabling a reduction to lower-degree cases via induction on the growth degree $d$. A combination theorem for rebuilding allows transfer from vertex stabilizers to the whole group, tying torsion growth vanishing to structural decompositions and Bass–Serre actions. Overall, the work links monodromy growth to torsion behavior, providing a robust framework for proving Lück-type equalities in a broad class of free-by-cyclic groups.

Abstract

We show that the homology torsion growth of a free-by-cyclic group with polynomially growing monodromy vanishes in every dimension independently of the choice of Farber chain. It follows that the integral torsion $ρ^\mathbb{Z}$ equals the $\ell^2$-torsion $ρ^{(2)}$ verifying a conjecture of Lück for these groups.

Theorems & Definitions (21)

• Conjecture 1.1: Lück
• Theorem A
• Corollary B
• Corollary C
• Conjecture 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof