Poisson structures on the Teichmueller space of hyperbolic surfaces with conical points

TL;DR

This paper compares two Poisson structures on the moduli space of hyperbolic surfaces with conical points, showing they are proportional under certain conditions and providing an explicit formula for one in terms of hyperbolic lengths.

Contribution

It establishes the proportionality of the Weil-Petersson and eta Poisson structures and derives an explicit formula for eta using hyperbolic lengths.

Findings

The two Poisson structures are multiples of each other when angles are pi.

An explicit formula for eta in terms of hyperbolic lengths is provided.

The proportionality holds for conical angles not exceeding 2pi.

Abstract

In this paper two Poisson structures on the moduli space of hyperbolic surfaces with conical points are compared: the Weil-Petersson one and the \eta coming from the representation variety. We show that they are multiple of each other, if the angles do not exceed 2\pi. Moreover, we exhibit an explicit formula for \eta in terms of hyperbolic lengths of a suitable system of arcs.