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Explicit Teichmüller curves with complementary series

Carlos MatheusGabriela Weitze-Schmithüsen — 2013

Bulletin de la Société Mathématique de France

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

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