Hausdorff dimension of divergent Teichmüller geodesics

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 $\theta$.