Rigidity properties of free-by-cyclic groups

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2. Subgroup separability and finite quotients

A group

$G$ is said to be locally extendable residually finite (LERF) if every finitely generated subgroup of

$G$ is an intersection of finite index subgroups of

$G$ .

Relatively hyperbolic case

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Conjecture 2.1.
If $G$ is a LERF relatively hyperbolic free-by-cyclic group which is not hyperbolic then $G$ is the fundamental group of a hyperbolic 3-manifold.
Hyperbolic case

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Conjecture 2.2.
If $G$ is a hyperbolic free-by-cyclic group with irreducible monodromy then $G$ is LERF if and only if the BNS invariant of every finite index subgroup of $G$ is symmetric.
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Problem 2.3.
Construct a hyperbolic free-by-cyclic group which is LERF.
Congruence subgroup property

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Problem 2.4.
[Tam Cheetham-West] Let $Q$ be a finite quotient of $\mathrm{Out}(F_n)$ for $n > 2$ . Does there exist a finite index characteristic subgroup $K$ of $F_n$ such that the quotient $\mathrm{Out}(F_n) \twoheadrightarrow Q$ factors the quotient $\mathrm{Out}(F_n) \twoheadrightarrow \mathrm{Out}(F_n /K)$ ?
Profinite invariants of free-by-cyclic groups

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Problem 2.5.
Which properties of free-by-cyclic groups are invariants of the profinite completion? How does the BNS invarian behave under profinite isomorphism of free-by-cyclic groups?

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