A 1/64 spectral gap for surfaces with $δ=1/2$

We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series $δ\in (\frac25,\frac{14}{27})$ has an essential spectral gap of size $β=\tfrac78(\tfrac12-δ)+\tfrac{1}{32}δ-ε$ for any $ε>0$. In particular, for $δ=\frac12$ this becomes $β=\tfrac{1}{64}-ε$. We show