Two-parametric hyperbolic octagons and reduced Teichmueller space in genus two

TL;DR

This paper models genus two Riemann surfaces using hyperbolic octagons with automorphisms, providing a detailed geometric and symplectic description of the associated Teichmüller space, with implications for quantum geometry and gravity.

Contribution

It introduces a two-parametric hyperbolic octagon model for genus two surfaces and explicitly describes the Teichmüller space using Fenchel-Nielsen coordinates and Weil-Petersson form.

Findings

Computed generators of the isometry group.

Provided a real-analytic parametrization of Teichmüller space.

Analyzed the structure of parameter space and isoperimetric orbits.

Abstract

It is explored a model of compact Riemann surfaces in genus two, represented geometrically by two-parametric hyperbolic octagons with an order four automorphism. We compute the generators of associated isometry group and give a real-analytic description of corresponding Teichm\"uller space, parametrized by the Fenchel-Nielsen variables, in terms of geometric data. We state the structure of parameter space by computing the Weil-Petersson symplectic two-form and analyzing the isoperimetric orbits. The results of the paper may be interesting due to their applications to the quantum geometry, chaotic systems and low-dimensional gravity.