Length spectrum characterization of asymptotic Teichmüller space
Let $f:S_1\to S_2$ be a quasiconformal mapping between two Riemann surfaces. The boundary length spectrum dilatation of $f$ is expressed by the following \[ L^*(f) = \inf_{E\subset S_1} \sup_{\gamma\subset S_1\setminus E} \max\left\{\frac{l_{S_2}f(\gamma)}{l_{S_1}}\right\}^{\pm1}, \] where the infimum runs over all compact sets $E$, the supremum is taken over homotopy classes of non-trivial closed curves $\gamma$ whose geodesic representative for the hyperbolic metric on $S_1$ is contained in the complement of $E$, and $l$ denotes the length of the geodesic representative for the corresponding hyperbolic metric. Suppose that $S_0$ admits an upper and lower bounded pair-of-pants decomposition, the paper under review shows that two marked surfaces $(S_1,f_1)$ and $(S_2,f_2)$ represent the same point in the asymptotic Teichmüller space $AT(S_0)$ if and only if $L^*(f_1\circ f_2^{-1})=1$, and then prove that the length spectrum metric on the asymptotic length spectrum Teichmüller space $AT_{ls}(S_0)$ defined by \[ d_{AL}(S_1,f_1,S_2,f_2) = \log L^*(f_1\circ f_2^{-1}) \] induces the same topology as the Teichmüller metric $d_{AT}$ on $AT(S_0)$. The paper also shows that the metrics are not bi-Lipschitz on $AT(S_0)$.
scientific article; zbMATH DE number 6871612
scientific article