Horospheres in Teichmüller space and mapping class group

Abstract

We study the geometry of horospheres in Teichmüller space of Riemann surfaces of genus g with n punctures, where $3g-3+n\ge 2$. We show that every $C^1$-diffeomorphism of Teichmüller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and metric balls, we obtain a new proof of Royden's Theorem that the isometry group of the Teichmüller metric is the extended mapping class group.

25 pages. Accepted for publication in ANNALES DE L'INSTITUT FOURIER