Combination of affine deformations on a hyperbolic surface

TL;DR

This paper explores how affine deformations of hyperbolic surfaces can be parametrized using invariants and twist parameters, linking Lorentzian geometry with Teichmüller theory, especially for Fenchel-Nielsen twists.

Contribution

It provides a new parametrization of affine deformation spaces using Margulis invariants and affine twists, connecting Lorentzian geometry with Fenchel-Nielsen coordinates.

Findings

Parametrization of affine deformation space using Margulis invariants and affine twists.

Explicit representation of tangent vectors on Teichmüller space via Lorentzian geometry.

Focus on Fenchel-Nielsen twists along separating geodesics.

Abstract

This paper is a continuation of the previous paper of the author[M]. We show that an affine deformation space of a hyperbolic surface of type (g,b) can be parametrized by Margulis invariants and affine twist parameters with a certain decomposition of the surface, which are associated with the Fenchel-Nielsen coordinates in Teichmuller theory. W.Goldman and G.Margulis[GM] introduced that a translation part of an affine deformation canonically corresponds to a tangent vector on the Teichmuller space. By this correspondence, we explicitly represent tangent vectors on the Teichmuller space from the perspective of Lorentzian geometry, only when the tangent vectors correspond to Fenchel-Nielsen twists along separating geodesic curves on a hyperbolic surface.