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4. Homological properties

    1. $\ell^2$-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.