Explicit Teichmüller curves with complementary series

Carlos Matheus; Gabriela Weitze-Schmithüsen

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 4, page 557-602
  • ISSN: 0037-9484

Abstract

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We construct an explicit family of arithmetic Teichmüller curves 𝒞 2 k , k ∈ ℕ , supporting SL ( 2 , ℝ ) -invariant probabilities μ 2 k such that the associated SL ( 2 , ℝ ) -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.

How to cite

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Matheus, Carlos, and Weitze-Schmithüsen, Gabriela. "Explicit Teichmüller curves with complementary series." Bulletin de la Société Mathématique de France 141.4 (2013): 557-602. <http://eudml.org/doc/272671>.

@article{Matheus2013,
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 $\mu _\{2k\}$ such that the associated $\textrm \{SL\}(2,\mathbb \{R\})$-representation on $L^2(\mathcal \{C\}_\{2k\}, \mu _\{2k\})$ has complementary series for every $k\ge 3$. 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 $\mathcal \{C\}_\{2k\}$ has arbitrarily slow rate of exponential mixing.},
author = {Matheus, Carlos, Weitze-Schmithüsen, Gabriela},
journal = {Bulletin de la Société Mathématique de France},
keywords = {moduli spaces; abelian differentials; translation surfaces; square-tiled surfaces; teichmüller curves; spectral gap; rate of mixing; complementary series},
language = {eng},
number = {4},
pages = {557-602},
publisher = {Société mathématique de France},
title = {Explicit Teichmüller curves with complementary series},
url = {http://eudml.org/doc/272671},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Matheus, Carlos
AU - Weitze-Schmithüsen, Gabriela
TI - Explicit Teichmüller curves with complementary series
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 4
SP - 557
EP - 602
AB - We construct an explicit family of arithmetic Teichmüller 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 $k\ge 3$. 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 $\mathcal {C}_{2k}$ has arbitrarily slow rate of exponential mixing.
LA - eng
KW - moduli spaces; abelian differentials; translation surfaces; square-tiled surfaces; teichmüller curves; spectral gap; rate of mixing; complementary series
UR - http://eudml.org/doc/272671
ER -

References

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