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1. Subgroups of free-by-cyclic groups

    1. Quasi-convex subgroups

          A subgroup $F$ of a group $G$ is a semi-fibre if there exists an injective non-surjective homomorphism $\varphi \colon F \to F$ such that $G$ is the HNN extension \[G = F \ast_{\varphi}.\] A subgroup $F$ of $G$ is a virtual semi-fibre if there exists a finite index subgroup $G'$ of $G$ such that $F$ is a semi-fibre of $G'$.

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      Conjecture 1.1.

      Let $G$ be a hyperbolic free-by-cyclic group and $H$ a non-quasiconvex subgroup of $G$. Then $H$ contains a subgroup $F$ which is a virtual fibre or a virtual semi-fibre of a free-by-cyclic subgroup of $G$.


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          Problem 1.2.

          [Naomi Andrew] What are the possible distortion functions for subgroups of free-by-cyclic groups?


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              Problem 1.3.

              [Yassine Guerch] If $G$ is a hyperbolic free-by-cyclic group then does $G$ contain a closed surface subgroup?

                • Effective coherence


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                  Problem 1.4.

                  [Ilya Kapovich] Given a finite set $S$ of elements of a free-by-cyclic group $G$, algorithmically find a presentation of the sugroup of $G$ generated by $S$.


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                      Problem 1.5.

                      Let $G$ be a hyperbolic free-by-cyclic groups. Which subgrups of $G$ admit Cannon–Thurston maps?


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                          Problem 1.6.

                          [Ilya Kapovich] Which subgroups of free-by-cyclic groups are Morse?

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