Explicit Teichmüller curves with complemetary series

Details Explicit Teichmüller curves with complemetary series Explicit Teichmüller curves with complemetary series

2011-09-02

arxiv.org/abs/1109.0517v1

Mathematics

Dynamical Systems

40 pages, 10 figures

Scientific paper

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

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