Rigidity properties of free-by-cyclic groups

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7. Miscellaneous

A group $G$ is said to be of type $VF$ if there exists a finite index subgroup $G'$ of $G$ which admits a finite classifying space.

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Conjecture 7.1.
[Naomi Andrew] If $G$ is a free-by-cyclic group then $\mathrm{Out}(G)$ is of type $VF$ .
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Problem 7.2.
Solve the subgroup membership problem for free-by-cyclic groups.
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Problem 7.3.
[Sam Hughes] Are free-by-cyclic groups linear? When do they embed in $\mathrm{SL}(2, \mathbb{C})$ ?
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Problem 7.4.
[Jean Pierre Mutanguha] Every outer automorphism $\phi \in \mathrm{Out}(F_n)$ admits a poset $P$ of attracting laminations. What is the length of the longest chain in $P$ ? Is the length an invariant of $G$ ?
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Problem 7.5.
[Edgar Bering] Develop a fibred face theory for four-manifolds which fibre over the circle with fibers handlebodies or connect sums of $\mathbb{S}^2 \times \mathbb{S}^1$ .
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Problem 7.6.
[Riley Lyman, Ilya Kapovich] Does there exist an analogue of pseudo-Anosov maps for homeomorphisms of handlebodies or connect sums of $\mathbb{S}^2 \times \mathbb{S}^1$ ?

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