A Uniform Strong Spectral Gap for Congruence Covers of a Compact Quotient of PSL(2, ℝ)d

The existence of a strong spectral gap for lattices in semi–simple Lie groups is crucial in many applications. In particular, for arithmetic lattices, it is useful to have bounds for the strong spectral gap that are uniform in the family of congruence covers. When the lattice is itself a congruence group, there are uniform and very good bounds for the spectral gap coming from the known bounds toward the Ramanujan–Selberg conjectures. In this note, we establish a uniform bound for the strong spectral gap for congruence covers of an irreducible co-compact lattice $\Gamma$ in ${\mathrm PSL}(2, \mathbb R)^d$ with $d\geq2$, which is the simplest and most basic case where the congruence subgroup property is not known.