  We provide the first information on diffeotopy groups of exotic smoothings of
R^4: For each of uncountably many smoothings, there are uncountably many
isotopy classes of self-diffeomorphisms. We realize these by various explicit
group actions. There are also actions at infinity by nonfinitely generated
groups, for which no nontrivial element extends over the whole manifold. In
contrast, every diffeomorphism of the end of the universal R^4 extends. Our
techniques apply to many other open 4-manifolds, and are related to cork
theory. We show that under broad hypotheses, cork twisting is equivalent (up to
blowups) to twisting on an exotic R^4, and give applications.
