Explicit Teichmüller curves with complementary series
Bulletin de la Société Mathématique de France, Volume 141 (2013) no. 4, p. 557-602

We construct an explicit family of arithmetic Teichmüller curves 𝒞 2k , k, supporting SL (2,)-invariant probabilities μ 2k such that the associated SL (2,)-representation on L 2 (𝒞 2k ,μ 2k ) has complementary series for every k3. 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 2k has arbitrarily slow rate of exponential mixing.

On construit une famille explicite de courbes de Teichmüller arithmétiques 𝒞 2k , k, supportant des probabilités SL (2,)-invariantes μ 2k telles que la SL (2,)-representation associée sur L 2 (𝒞 2k ,μ 2k ) a des séries complémentaires pour tout k3. 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 2k a une vitesse de mélange exponentiel arbitrarement lente.

DOI : https://doi.org/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
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {141},
     number = {4},
     year = {2013},
     pages = {557-602},
     doi = {10.24033/bsmf.2656},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2013__141_4_557_0}
}
Explicit Teichmüller curves with complementary series. Bulletin de la Société Mathématique de France, Volume 141 (2013) no. 4, pp. 557-602. doi : 10.24033/bsmf.2656. http://www.numdam.org/item/BSMF_2013__141_4_557_0/

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