  Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$,
whose law is invariant under isometries of $\H^2$. We prove that if the
probability that $Z$ contains a fixed ball of radius 1 is larger than some
universal constant $p<1$, then there is positive probability that $Z$ contains
(bi-infinite) lines.
  We then consider a family of random sets in $\H^2$ that satisfy some
additional natural assumptions. An example of such a set is the covered region
in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment
of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays
fast enough, then there are almost surely no lines in $Z$. We also show that if
the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$.
In the case of the Poisson Boolean model with balls of fixed radius $R$ we
characterize the critical intensity for the almost sure existence of lines in
the covered region by an integral equation.
  We also determine when there are lines in the complement of a Poisson process
on the Grassmannian of lines in $\H^2$.
