A Hyperelliptic View on Teichmuller Space. II

TL;DR

This paper demonstrates that the Teichmuller space of all closed Riemann surfaces can be viewed as fibered over hyperelliptic surfaces, with algebraic fiber bundles, and offers an elementary proof of Toledo's rigidity theorem.

Contribution

It establishes a fibering of Teichmuller space over hyperelliptic surfaces and provides a simplified proof of Toledo's rigidity theorem.

Findings

Teichmuller space is fibered twice over hyperelliptic Teichmuller space.

The fiber bundles are real algebraic and define an embedding into a product space.

Provides an elementary proof of Toledo's rigidity theorem.

Abstract

Using the methods of the previous paper [ABG], we show that the Teichmuller space T of all closed Riemann surfaces is fibred twice over the Teichmuller space H of hyperelliptic ones. Both fibre bundles \pi_1,\pi_2:T->H are real algebraic (rational). They define an embedding T->HxH. In addition, we indicate slight modifications of the proof of [ABG, Theorem 5.1] providing an elementary proof of Toledo's rigidity theorem.