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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.