Horospheres in Teichm\"uller space and mapping class group

We study the geometry of horospheres in Teichm"uller space of Riemann surfaces of genus g with n punctures, where $3g-3+n\ge 2$. We show that every $C^1$-diffeomorphism of Teichm"uller 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 Teichm"uller metric is the extended mapping class group.