Abstract
Let \(\mathcal {C}\) be a connected component of a stratum of the moduli space of unit area translation surfaces of genus \(g\ge 1\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Sect. 3.4.
- 2.
In the language of unitary \(SL(2,\mathbb {R})\)-representations (see Sect. 3.4), the precise statement of Yoccoz’s conjecture is: “the regular \(SL(2,\mathbb {R})\) representation \(L^2(\mathcal {C},\mu _{\mathcal {C}})\) has no complementary series whenever \(\mu _{\mathcal {C}}\) is a Masur-Veech measure”. So far, the validity of this conjecture is known only for the moduli space of unit area flat torii \(\mathcal {C}=SL(2,\mathbb {R})/SL(2,\mathbb {Z})\) thanks to a classical theorem of Selberg.
- 3.
Selberg used cyclic covers to show that there is no uniform spectral gap for the Laplacian of \(\mathbb {H}/\Gamma \) (where \(\Gamma \subset SL(2,\mathbb {Z})\) is a lattice). As it turns out, this is equivalent to our assertion on rates of mixing: see Sect. 3.4.
- 4.
I.e., the tangent space of \(SL(2,\mathbb {R})\) at the identity.
- 5.
I.e., \([A, B]:= AB-BA\) is the commutator.
- 6.
That is, \(\langle \Omega _{\rho }v, w\rangle = \langle v,\Omega _{\rho }w\rangle \) for any \(C^2\)-vectors \(v, w\in \mathcal {H}\).
- 7.
The original arguments of Ratner allow one to explicitly these constants: see our paper [48] for more details.
- 8.
This is more clearly seen when f and g are characteristic functions of Borelian sets.
- 9.
This proposition fails for \(N<6\): indeed, \(\mathbb {H}/\Gamma _n\) has genus 0 for \(1\le n\le 5\).
- 10.
Because \(\Gamma _6\) has index 72 in \(SL(2,\mathbb {Z})\) and the fundamental domain \(\mathcal {F}_1=\{z\in \mathbb {H}: |\text {Re}(z)|\le 1/2, |z|\ge 1\}\) of \(\mathbb {H}/SL(2,\mathbb {Z})\) has hyperbolic area \(\int _{-1/2}^{1/2}\int _{\sqrt{1-x^2}}^{\infty }\frac{dxdy}{y^2}=\pi /3\).
- 11.
\(\Gamma \subset SL(2,\mathbb {Z})\) is a congruence subgroup if \(\Gamma \) contains the principal congruence subgroup \(\Gamma _N\) for some \(N\in \mathbb {N}\).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Matheus Silva Santos, C. (2018). Arithmetic Teichmüller Curves with Complementary Series. In: Dynamical Aspects of Teichmüller Theory. Atlantis Studies in Dynamical Systems, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-92159-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-92159-4_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92158-7
Online ISBN: 978-3-319-92159-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)