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We study the geometry of horospheres in Teichm\"uller space of Riemannsurfaces of genus g with n punctures, where $3g-3+n\geq 2$. We show that every$C^1$-diffeomorphism of Teichm\"uller space to itself that preserveshorospheres is an element of the extended mapping class group. Using therelation between horospheres and metric balls, we obtain a new proof ofRoyden's Theorem that the isometry group of the Teichm\"uller metric is theextended mapping class group.

Horospheres in Teichm\"uller space and mapping class group