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Hausdorff dimension of divergent Teichmüller geodesics


Author: Howard Masur
Journal: Trans. Amer. Math. Soc. 324 (1991), 235-254
MSC: Primary 30F30; Secondary 28A75, 32G15, 58F17
DOI: https://doi.org/10.1090/S0002-9947-1991-0984857-0
MathSciNet review: 984857
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Abstract: Let $ g > 1$ be given and let $ k = ({k_1}, \ldots ,{k_n})$ be an $ n$-tuple of positive integers whose sum is $ 4g - 4$. Denote by $ {Q_k}$ the set of all holomorphic quadratic differentials on compact Riemann surfaces of genus $ g$ whose zeros have orders $ {k_1}, \ldots, {k_n}$. $ Q_k$ is called a stratum inside the cotangent space of all holomorphic quadratic differentials over the Teichmüller space of genus $ g$. Let $ {Q_k}/\operatorname{Mod} (g)$ be the moduli space where $ \operatorname{Mod} (g)$ is the mapping class group. Each $ q \in {Q_k}$ defines a Teichmüller geodesic.

Theorem. There exists $ \delta > 0$ so that for almost all $ q \in {Q_k}$, the set of $ \theta$, such that the geodesic defined by $ {e^{i\theta }}q$ eventually leaves every compact set in $ {Q_k}/\operatorname{Mod} (g)$, has Hausdorff dimension $ \theta$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-0984857-0
Article copyright: © Copyright 1991 American Mathematical Society

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