Complex Fenchel-Nielsen coordinates with small imaginary parts

TL;DR

This paper provides a new proof that nearly real complex Fenchel-Nielsen coordinates induce quasiFuchsian representations, strengthening previous results and connecting to hyperbolic 3-manifold dynamics.

Contribution

It offers a refined proof of a key theorem relating Fenchel-Nielsen coordinates to quasiFuchsian representations, with improved conditions and applications to hyperbolic 3-manifold theory.

Findings

New proof of Fenchel-Nielsen coordinate theorem

Stronger conditions on coordinate near-realness

Application of exponential mixing in hyperbolic geometry

Abstract

Kahn and Markovic \cite{KahnMark} proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a theorem which states that an assignment of nearly real, complex Fenchel-Nielsen coordinates to the cuffs of a pants decomposition of a closed surface $S$ induces a quasiFuchsian representation of the fundamental group of $S$. We give a new proof of this theorem with a slightly stronger conditions on the Fenchel-Nielsen coordinates and explain how to use the exponential mixing of the geodesic flow on a closed hyperbolic three manifold to prove that our theorem is sufficient for the applications in the work of Kahn and Markovic \cite{KahnMark}.