Torsion homology growth of polynomially growing free-by-cyclic groups (Q6635520)

Several mathematicians have studied the notions related to homology growth for CW-complexes, such as Betti numbers, minimal numbers of generators of singular homology, order of singular homology torsion, torsion invariants etc. In 2013, \textit{W. Lück} [Geom. Funct. Anal. 23, No. 2, 622--663 (2013; Zbl 1273.22009)] conjectured an approximation theorem for the \(\ell^2\)-torsion and the integral torsion for infinite residually finite \(\ell^2\)-acyclic group of type VF. He also gave some evidence towards the conjecture by proving it for fundamental groups of closed aspherical manifolds which admit nontrivial \(S^1\)-action.\N\NIn the paper under review, the authors verify this conjecture for groups \(G = F^n \rtimes_{\phi} \mathbb{Z}\) which are (finitely generated) free-by-(infinite)cyclic \(\langle\phi\rangle\), where \(\phi\) is polynomially growing. The main tool to compute the homology torsion growth is something called the cheap rebuilding property introduced in a recent important paper [``On homology torsion growth'', Preprint, \url{arXiv:2106.13051}] by \textit{M. Abert} et al.