  We give a way of estimating quickly the length of the hyperbolic isometry
associated to a plane Cremona transformation that is presented as the product
of two involutions. We also show, using the ideas of Cantat and Lamy, that,
provided that the ground field is either algebraic of characteristic not 2 or
of characteristic zero, the normal closure of a high power of such a
transformation is a proper subgroup of the Cremona group; this gives an
effective instantiation of some of their results. Moreover, we show that any
hyperbolic Cremona transformation is rigid and give a simple criterion for it
to be tight.
  This version is thoroughly revised from the previous version. In particular,
it corrects an error in an earlier version that was pointed out by Lonjou.
