  The out-of-plane hopping in layered metals with strong electronic correlation
is analyzed theoretically. By studying the effects of in-plane interactions on
the interlayer tunnelling we investigate one of the oldest unresolved problems
of high-T$_c$ cuprates, the so-called bilayer splitting. The strong
electron-electron interactions within each layer reduce the quasiparticle
weight, and modify the hopping between layers. We analyze the effect of the
in-plane correlations on the interlayer hopping using a Renormalization Group
scheme already applied to the problem of interlayer coherence . The bilayer
band splitting acquires a significant temperature and doping dependence, and
can be completely suppressed when the Fermi energy coincides with a Van Hove
singularity.
