Horospheres in Teichm\"uller space and mapping class group
Weixu Su, Dong Tan

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 . We show that every -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.
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Taxonomy
TopicsAnalytic and geometric function theory · Geometric and Algebraic Topology · Holomorphic and Operator Theory
