No Arabic abstract
We introduce a function model for the Teichmuller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichmuller space. We prove that the identity map from the Teichmuller space equipped with the usual Teichmuller metric to the Teichmuller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichmuller space equipped with this new metric to the Teichmuller space equipped with the usual Teichmuller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichmuller metric on the Teichmuller space. We give a remark about the pressure metric and the Weil-Petersson metric.
We construct an Ahlfors-Bers complex analytic model for the Teichmuller space of the universal hyperbolic lamination (also known as Sullivans Teichmuller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivans Teichmuller space is Kahler isometric biholomorphic to the space of continuous functions from the profinite completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichmuller space of this surface; i.e. We find natural Kahler coordinates for the Sullivans Teichmuller space. This is the main result. As a corollary, we show the expected fact that the Nag-Verjovsky embedding is transversal to the Sullivans Teichmuller space contained in the universal one.
We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.
This article gives a complex analysis lighting on the problem which consists in restoring a bordered connected riemaniann surface from its boundary and its Dirichlet-Neumann operator. The three aspects of this problem, unicity, reconstruction and characterization are approached.
We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichm{u}ller geodesic flow.
We investigate the translation lengths of group elements that arise in random walks on weakly hyperbolic groups. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov and almost every random walk on $mathrm{Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk. We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmuller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.