Orbits in Teichmüller dynamics admits a critical exponent gap

McMullen '03 constructs a collection of orbits \(\mathrm{SL}_2(\mathbb{R}).x\) in \(\mathcal{H}(1,1)\) with infinitely generated stabilizers \(\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)\). We prove a gap in the set of critical exponents of stabilizers of \(\mathrm{SL}_2(\mathbb{R})\)-orbits in \(\mathcal{H}_g\): for every \(x\in \mathcal{H}_g\), either \(\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)\) is a lattice, or we have a uniform bound on the critical exponent \(\delta(\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)) \le 1-\varepsilon_g\).