Horospheres in Teichm\"uller space and mapping class group

TL;DR

This paper investigates the geometry of horospheres in Teichmüller space and demonstrates that any smooth symmetry preserving horospheres must be an extended mapping class group element, providing a new proof of Royden's Theorem.

Contribution

It establishes that horosphere-preserving diffeomorphisms are exactly the extended mapping class group elements, offering a novel proof of Royden's Theorem.

Findings

Horospheres characterize the extended mapping class group.

Any $C^1$-diffeomorphism preserving horospheres is an extended mapping class group element.

Provides a new proof of Royden's Theorem using horospheres and metric balls.

Abstract

We study the geometry of horospheres in Teichm\"uller space of Riemann surfaces 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 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.