  We establish four results concerning connections between actions on separable
C*-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra
Z. For actions of residually finite groups or of the reals which have finite
Rokhlin dimension with commuting towers, we show that if the action of any
nontrivial group element is approximately inner then the C*-algebra acted upon
is Z-stable. Without the assumption on approximate innerness, we show that the
crossed product has good divisibility properties under mild assumptions. We
also establish an analogous result for the generalized tracial Rokhlin property
and tracial versions of approximate innerness and Z-absorption for actions of
finite groups and of the integers. For actions of a single automorphism which
have the Rokhlin property, we show that a condition which is strictly weaker
than requiring that some power of the automorphism is approximately inner is
sufficient to obtain that the crossed product absorbs Z even when the original
algebra is not.
