Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

TL;DR

This paper introduces a generalized class of cluster mutations related to Riemann surfaces with orbifold points, providing new combinatorial and algebraic tools for understanding their Teichmüller spaces.

Contribution

It extends cluster mutation theory to include reciprocal polynomial transformations and applies this to describe Teichmüller spaces of orbifold Riemann surfaces with new combinatorial and algebraic frameworks.

Findings

Defined new cluster mutations with reciprocal polynomial exchange transformations.

Constructed the Poisson algebra for Teichmüller space coordinates.

Developed a combinatorial description of geodesic functions and mapping class group actions.

Abstract

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_o$. We propose the dual graph description of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the Teichm\"uller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.