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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.


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          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?
              Suppose that H is virtually RFRS and quasi-isometric to a free-by-cyclic group G. Then H is virtually free-by-cyclic.


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              Problem 3.3.

              Construct two hyperbolic irreducible free-by-cyclic groups which are not quasi-isometric.


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                  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.


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                      Problem 3.5.

                      [Yassine Guerch] What are the measure equivalence classes of free-by-cyclic groups?


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                          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.