Orbits in Teichm\"uller dynamics admits a critical exponent gap

TL;DR

This paper proves a gap in the critical exponents of stabilizers of SL(2,R)-orbits in Teichmüller space, showing they are either lattices or have exponents bounded away from 1.

Contribution

It establishes a uniform gap in the critical exponents of stabilizers, extending McMullen's construction to a broader setting in Teichmüller dynamics.

Findings

Stabilizers are either lattices or have critical exponents bounded by 1 - epsilon_g.

A uniform bound exists for the critical exponents of non-lattice stabilizers.

The result applies to all points in the moduli space alg.

Abstract

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 $\delta(\mathrm{stab}_{\mathrm{SL}_2(\mathbb{R})}(x)) \le 1-\varepsilon_g$.