  We consider locally conformal Kaehler geometry as an equivariant (homothetic)
Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a
pair (K,\Gamma) where K is a Kaehler manifold and \Gamma a discrete Lie group
of biholomorphic homotheties acting freely and properly discontinuously.
  We define a new invariant of a locally conformal Kaehler manifold (K,\Gamma)
as the rank of a natural quotient of \Gamma, and prove its invariance under
reduction. This equivariant point of view leads to a proof that locally
conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman
manifolds and is equivalent to a Sasakian reduction.
  Moreover we define locally conformal hyperkaehler reduction as an equivariant
version of hyperkaehler reduction and in the compact case we show its
equivalence with 3-Sasakian reduction. Finally we show that locally conformal
hyperkaehler reduction induces hyperkaehler with torsion (HKT) reduction of the
associated HKT structure and the two reductions are compatible, even though not
every HKT reduction comes from a locally conformal hyperkaehler reduction.
