The intrinsic asymmetry and inhomogeneity of Teichmuller space

TL;DR

This paper extends Royden's theorems to a broader class of metrics on Teichmuller space, revealing intrinsic asymmetry and inhomogeneity, and providing new insights into the structure of the space.

Contribution

It generalizes Royden's results from the Teichmuller metric to arbitrary invariant Finsler metrics, highlighting the space's intrinsic asymmetry.

Findings

Teichmuller space exhibits intrinsic asymmetry under various metrics

Royden's theorems can be extended to Finsler metrics beyond Teichmuller metric

New mechanisms explain the structure and symmetries of Teichmuller space

Abstract

Royden proved that any isometry of Teichmuller space in the Teichmuller metric must be an element of the extended mapping class group M(S). He also proved that the Teichmuller metric is not symmetric at any point. In this paper we give extensions of Royden's theorems from the Teichmuller metric to an arbitrary complete, finite covolume, M(S)-invariant Finsler (e.g. Riemannian) metric on Teichmuller space. In particular this gives a new mechanism behind Royden's original theorem.