Rigidity properties of free-by-cyclic groups

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4. Homological properties

$\ell^2$ -torsion

Clay shows that $\ell^2$ -torsion vanishes for free-by-cyclic groups with polynomially-growing monodromies [MR3667215].

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Conjecture 4.1.
[Matt Clay] If $G$ is a free-by-cyclic group with exponentially growing monodromy then it has non-vanishing $\ell^2$ -torsion.
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Problem 4.2.
[Sam Hughes] If $G$ is a free-by-cyclic group with exponentially growing monodromy then does it have non-vashing homology torsion growth?
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Problem 4.3.
[Tam Cheetham-West] Let $G$ be a non-geometric free-by-cyclic group and $A$ a finitely generated abelian group. Does there exist a finite index subgroup $G'$ of $G$ such that $A$ is isomorphic to a direct summand of the abelianisation of $G'$ ?
RFRS

A group $G$ is residually finite rationally solvable (RFRS) if there exists a non-ascending residual chain $(H_i)_{i \in \mathbb{N}}$ of finite index normal subgroups $N_i \trianglelefteq G$ with $H_0 = G$ , such that $\mathrm{ker}\, \alpha_i \leq H_{i+1}$ for every $i$ , where $\alpha_i \colon H_i \to H_i^{\mathrm{fab}}$ is the free abelianisation map.

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Problem 4.4.
Characterise the non-hyperbolic free-by-cyclic groups which are virtually RFRS.
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Problem 4.5.
[Sam Hughes] Are all free-by-cyclic groups virtually (residually torsion-free nilpotent)?
1. Remark. The property of being residually torsion-free nilpotent is weaker than that of being RFRS.

Cite this as: AimPL: Rigidity properties of free-by-cyclic groups, available at http://aimpl.org/freebycyclic.