3. Quasi-isometric and measure equivalence rigidity
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Conjecture 3.1.
[Jean Pierre Mutanguha] Let $G$ and $H$ be free-by-cyclic groups which are quasi-isometric. Then $G$ admits irreducible and atoroidal monodromy if and only if $H$ admits irreducible and atoroidal monodromy. -
Suppose that $H$ is virtually RFRS and quasi-isometric to a free-by-cyclic group $G$. Then H is virtually free-by-cyclic.
Problem 3.2.
[Genevieve Walsh] If a group $G$ is quasi-isometric to a free-by-cyclic group then is it true that $G$ is virtually free-by-cyclic? -
Problem 3.3.
Construct two hyperbolic irreducible free-by-cyclic groups which are not quasi-isometric. -
Conjecture 3.4.
[Jean Pierre Mutanguha] Let $F = F(a,b,c,d,e)$ be a free group of rank five. Define an automorphism $\phi \in \mathrm{Aut}(F)$ such that \[\begin{split} \phi \colon F(a,b,c,d,e) &\to F(a,b,c,d,e) \\ a &\mapsto b\\ b &\mapsto c\\ c &\mapsto ab\\ d &\mapsto ea \\ e &\mapsto ed \end{split}\] Let $F'$ be the free factor of $F$ generated by $\{a,b,c\}$. Let $G = F \rtimes_{\phi} \mathbb{Z}$ be the mapping torus of $\phi$, and let $\Gamma$ be the mapping torus of the restriction of $\phi$ to $F'$. Then $\Gamma$ and $G$ are not quasi-isometric. -
Problem 3.5.
[Yassine Guerch] What are the measure equivalence classes of free-by-cyclic groups? -
Conjecture 3.6.
[Chris Leininger] Let $G$ and $H$ be relatively hyperbolic non-hyperbolic free-by-cyclic groups with a trivial JSJ decomposition. If $G$ and $H$ are quasi-isometric then they are commensurable.
Cite this as: AimPL: Rigidity properties of free-by-cyclic groups, available at http://aimpl.org/freebycyclic.