The essential spectral gap conjecture for convex co-compact groups

Let $X$ be the convex co-compact hyperbolic surface associated with a convex co-compact group $\Lambda$, let $\delta$ denote the Hausdorff dimension of its limit set, let $R(s)$ be the meromorphically continued resolvent, and let ${\mathcal R}_X$ be its resonance set. Define

$$G(\Lambda):=\inf\left\{\sigma<\delta:\ \{\operatorname{Re}(s)\geq\sigma\}\cap{\mathcal R}_X\text{ is finite}\right\}.$$

Essential spectral gap conjecture. One has

$$G(\Lambda)=\frac{\delta}{2}.$$

The quantity $G(\Lambda)$ measures the location of the essential spectral gap, or equivalently the largest real part at which resonances can accumulate. Earlier results give $G(\Lambda)\leq\frac12$ when $\delta>\frac12$ and $G(\Lambda)<\delta$ when $\delta\leq\frac12$; the conjectured exact value is not established in the supplied text.