  Motivated by the study of the conjugacy problem for outer automorphism of
free groups, we develop the algorithmic theory of the free-by-cyclic group
produced by unipotent linearly growing automorphisms of f.g. free groups.
  We compute canonical splittings of these suspensions as well as their
subgroups. We compute their automorphism groups. We show that this class of
suspensions is effectively coherent. We solve the mixed Whitehead problem in
these suspensions and show that their subgroups all satisfy the Minkowski
property, i.e. that torsion in their outer automorphism group is faithfully
represented in some computable finite quotients.
  An application of our results is a solution to the conjugacy problem for
outer automorphisms of free groups whose polynomially growing part is unipotent
linear.
