A generalized Baumslag–Solitar group (GBS group) is a finitely generated group G
which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that
Out(G) either contains non-abelian free groups or is virtually nilpotent of class ≤2. It
has torsion only at finitely many primes.
One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it
is, one may decide whether it is virtually abelian, or finitely generated. The
isomorphism problem is solvable among GBS groups with Out(G) virtually
If G is unimodular (virtually Fn×Z), then Out(G) is commensurable with a
semi-direct product Zk⋊Out(H) with H virtually free.