  For all convex co-compact hyperbolic surfaces, we prove the existence of an
essential spectral gap, that is a strip beyond the unitarity axis in which the
Selberg zeta function has only finitely many zeroes. We make no assumption on
the dimension $\delta$ of the limit set, in particular we do not require the
pressure condition $\delta\leq {1\over 2}$. This is the first result of this
kind for quantum Hamiltonians.
  Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589].
The main new ingredient is the fractal uncertainty principle for
$\delta$-regular sets with $\delta<1$, which may be of independent interest.
