  For a finitely generated regular wreath product, the metric is known, but its
computation can be an NP-complete problem. Also, it is not known for the
nonregular case.
  In this article, a metric estimate is defined for regular wreath products
which can be computed in polynomial time, based on the metrics of the factors.
This estimate is then used to study the distortion of some natural subgroups of
a wreath product. Finally, the metric estimate is generalized to the nonregular
case.
