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arXiv · 1109.0517

Explicit Teichmüller curves with complemetary series

Carlos Matheus·Gabriela Weitze-Schmithüsen
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Abstract

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

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Carlos Matheus, Gabriela Weitze-Schmithüsen. 2012-08-22. Explicit Teichmüller curves with complemetary series. https://arxiv.org/abs/1109.0517

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