  In this work we study the Poisson Boolean model of percolation in locally
compact Polish metric spaces and we prove the invariance of subcritical and
supercritical phases under mm-quasi-isometries. In other words, we prove that
if the Poisson Boolean model of percolation is subcritical or supercritical (or
exhibits phase transition) in a metric space M which is mm-quasi-isometric to a
metric space N, then these phases also exist for the Poisson Boolean model of
percolation in N. Then we apply these results to understand the phenomenon of
phase transition in a large family of metric spaces. Indeed, we study the
Poisson Boolean model of percolation in the context of Riemannian manifolds, in
a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the
existence of a subcritical phase in Gromov spaces with bounded growth at some
scale.
