Explicit Teichmüller curves with complementary series

Matheus, Carlos; Weitze-Schmithüsen, Gabriela

doi:10.24033/bsmf.2656

Explicit Teichmüller curves with complementary series
[Courbes de Teichmüller explicites avec série complémentaires]

Matheus, Carlos ; Weitze-Schmithüsen, Gabriela

Bulletin de la Société Mathématique de France, Tome 141 (2013) no. 4, pp. 557-602.

Résumé
Abstract

On construit une famille explicite de courbes de Teichmüller arithmétiques $𝒞_{2 k}$ , $k \in ℕ$ , supportant des probabilités $SL (2, ℝ)$ -invariantes $μ_{2 k}$ telles que la $SL (2, ℝ)$ -representation associée sur $L^{2} (𝒞_{2 k}, μ_{2 k})$ a des séries complémentaires pour tout $k \geq 3$ . En fait, la taille du trou spectral de cette famille tend vers zéro. En particulier, le flot géodésique de Teichmüller restreint à ces courbes de Teichmüller explicites $𝒞_{2 k}$ a une vitesse de mélange exponentiel arbitrarement lente.

We construct an explicit family of arithmetic Teichmüller curves $𝒞_{2 k}$ , $k \in ℕ$ , supporting $SL (2, ℝ)$ -invariant probabilities $μ_{2 k}$ such that the associated $SL (2, ℝ)$ -representation on $L^{2} (𝒞_{2 k}, μ_{2 k})$ has complementary series for every $k \geq 3$ . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves $𝒞_{2 k}$ has arbitrarily slow rate of exponential mixing.

DOI : 10.24033/bsmf.2656

Classification : 37D40
Keywords: moduli spaces, abelian differentials, translation surfaces, square-tiled surfaces, teichmüller curves, spectral gap, rate of mixing, complementary series
Mot clés : espaces de modules, différentielles abéliennes, surfaces de translation, surfaces à petits carreaux, courbes de teichmüller, trou spectral, vitesse de mélange, série complémentaire

@article{BSMF_2013__141_4_557_0,
     author = {Matheus, Carlos and Weitze-Schmith\"usen, Gabriela},
     title = {Explicit {Teichm\"uller} curves with complementary series},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {557--602},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {141},
     number = {4},
     year = {2013},
     doi = {10.24033/bsmf.2656},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/bsmf.2656/}
}

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Matheus, Carlos; Weitze-Schmithüsen, Gabriela. Explicit Teichmüller curves with complementary series. Bulletin de la Société Mathématique de France, Tome 141 (2013) no. 4, pp. 557-602. doi : 10.24033/bsmf.2656. http://www.numdam.org/articles/10.24033/bsmf.2656/

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