  We construct minimal laminations by hyperbolic surfaces whose generic leaf is
a disk and contain any prescribed family of surfaces and with a precise control
of the topologies of the surfaces that appear. The laminations are constructed
via towers of finite coverings of surfaces for which we need to develop a
relative version of residual finiteness which may be of independent interest.
The main step in establishing this relative version of residual finiteness is
to obtain finite covers with control on the \emph{second systole} of the
surface, which is done in the appendix. In a companion paper, the case of other
generic leaves is treated.

Geometric Topology;Dynamical Systems