Explicit Teichmüller curves with complementary series

Matheus, Carlos; Weitze-Schmithüsen, Gabriela

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

We construct an explicit family of arithmetic Teichmüller curves ${𝒞}_{2k}$, $k\in \mathbb{N}$, supporting $\mathrm{SL}(2,ℝ)$-invariant probabilities ${\mu }_{2k}$ such that the associated $\mathrm{SL}(2,ℝ)$-representation on $L^2({𝒞}_{2k},{\mu }_{2k})$ has complementary series for every $k\ge 3$. 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 ${𝒞}_{2k}$ has arbitrarily slow rate of exponential mixing.

On construit une famille explicite de courbes de Teichmüller arithmétiques ${𝒞}_{2k}$, $k\in \mathbb{N}$, supportant des probabilités $\mathrm{SL}(2,ℝ)$-invariantes ${\mu }_{2k}$ telles que la $\mathrm{SL}(2,ℝ)$-representation associée sur $L^2({𝒞}_{2k},{\mu }_{2k})$ a des séries complémentaires pour tout $k\ge 3$. 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 ${𝒞}_{2k}$ a une vitesse de mélange exponentiel arbitrarement lente.

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

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