  The aim of this work is to understand some of the asymptotic properties of
sequences of lattices in a fixed locally compact group. In particular we will
study the asymptotic growth of the Betti numbers of the lattices renormalized
by the covolume and the rank gradient, the minimal number of generators also
renormalized by the covolume. For doing so we will consider the ultraproduct of
the sequence of actions of the locally compact group on the coset spaces and we
will show how the properties of one of its cross sections are related to the
asymptotic properties of the lattices.
