  We study the reduced Donaldson-Thomas theory of abelian threefolds using
Bridgeland stability conditions. The main result is the invariance of the
reduced Donaldson-Thomas invariants under all derived autoequivalences, up to
explicitly given wall-crossing terms. We also present a numerical criterion for
the absence of walls in terms of a discriminant function. For principally
polarized abelian threefolds of Picard rank one, the wall-crossing
contributions are discussed in detail. The discussion yield evidence for a
conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin,
and the first author.
  For the proof we strengthen several known results on Bridgeland stability
conditions of abelian threefolds. We show that certain previously constructed
stability conditions satisfy the full support property. In particular, the
stability manifold is non-empty. We also prove the existence of a Gieseker
chamber and determine all wall-crossing contributions. A definition of reduced
generalized Donaldson-Thomas invariants for arbitrary Calabi-Yau threefolds
with abelian actions is given.
