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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

5. Geometry of free-by-cyclic groups


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      Problem 5.1.

      Characterise the free-by-cyclic groups which act geometrically on CAT(0) spaces.


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          Problem 5.2.

          [Rylee Lyman] Let $F = F(a,b,c)$ be a free group of rank 3. Define an automorphism $\phi \in \mathrm{Aut}(F)$ such that \[ \begin{split} \phi \colon F(a,b,c) &\to F(a,b,c) \\ a &\mapsto a \\ b &\mapsto ab \\ c &\mapsto bcb. \end{split} \] Does $G = F \rtimes_{\phi} \mathbb{Z}$ act geometrically on a CAT(0) cube complex?
            1. Remark. It is known that the group $G$ acts geometrically on a CAT(0) space.


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                  Problem 5.3.

                  What’s the relationship between being CAT(0) and RFRS for free-by-cyclic groups?

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