  We prove that the homology groups of a principal ample groupoid vanish in
dimensions greater than the dynamic asymptotic dimension of the groupoid (as a
side-effect of our methods, we also give a new model of groupoid homology in
terms of the Tor groups of homological algebra, which might be of independent
interest). As a consequence, the K-theory of the $C^*$-algebras associated with
groupoids of finite dynamic asymptotic dimension can be computed from the
homology of the underlying groupoid. In particular, principal ample groupoids
with dynamic asymptotic dimension at most two and finitely generated second
homology satisfy Matui's HK-conjecture.
  We also construct explicit maps from the groupoid homology groups to the
K-theory groups of their $C^*$-algebras in degrees zero and one, and
investigate their properties.
