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arXiv · 2411.09144

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

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

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Omri Nisan Solan. 2024-11-14. Orbits in Teichm\"uller dynamics admits a critical exponent gap. https://arxiv.org/abs/2411.09144

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