  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
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 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.
