  We study the mapping class group of a nontrivial irreducible shift of finite
type: the group of flow equivalences of its mapping torus modulo isotopy. This
group plays for flow equivalence the role that the automorphism group plays for
conjugacy. It is countable; not residually finite; acts faithfully (and
n-transitively, for all n) by permutations on the set of circles in the mapping
torus; has solvable word problem and trivial center; etc. There are many open
problems.
