  We study the automorphism groups of free-by-cyclic groups and show these are
finitely generated in the following cases: (i) when defining automorphism has
linear growth and (ii) when the rank of the underlying free group has rank at
most 3.
  The techniques we use are actions on trees, including the trees of cylinders
due to Guirardel and Levitt, the relative hyperbolicity of free-by-cyclic
groups (due to Gautero and Lustig, Ghosh, and Dahmani and Li) and the
filtration of the automorphisms of a group preserving a tree, following Bass
and Jiang, and Levitt.
  Our general strategy is to produce an invariant tree for the group and study
that, usually reducing the initial problem to some sort of McCool problem (the
study of an automorphism group fixing some collection of conjugacy classes of
subgroups) for a group of lower complexity. The obstruction to pushing these
techniques further, inductively, is in finding a suitable invariant tree and in
showing that the relevant McCool groups are finitely generated.
