  The existence of a strong spectral gap for quotients $\Gamma\bs G$ of
noncompact connected semisimple Lie groups is crucial in many applications. For
congruence lattices there are uniform and very good bounds for the spectral gap
coming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$
has no compact factors then for general lattices a strong spectral gap can
still be established, however, there is no uniformity and no effective bounds
are known. This note is concerned with the strong spectral gap for an
irreducible co-compact lattice $\Gamma$ in $G=\PSL(2,\bbR)^d$ for $d\geq 2$
which is the simplest and most basic case where the congruence subgroup
property is not known. The method used here gives effective bounds for the
spectral gap in this setting.
