Conformal grafting and convergence of Fenchel-Nielsen twist coordinates

TL;DR

This paper investigates how Fenchel-Nielsen twist coordinates behave when hyperbolic surfaces are modified by attaching cylinders with increasing moduli, showing convergence of these coordinates in the limit.

Contribution

It establishes the convergence of Fenchel-Nielsen twist coordinates for hyperbolic surfaces after grafting with cylinders of unbounded moduli, providing new insights into surface deformation.

Findings

Fenchel-Nielsen twist coordinates converge as cylinder moduli tend to infinity.

Grafting along simple closed curves affects the twist parameters predictably.

The results deepen understanding of hyperbolic surface deformations and moduli space behavior.

Abstract

We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.