Hyperbolic 3-manifolds and Cluster Algebras

TL;DR

This paper explores the application of cluster algebras and y-variables to understand hyperbolic 3-manifolds, particularly in solving edge-gluing conditions and analyzing hyperbolic structures on mapping tori.

Contribution

It introduces a novel approach using cluster y-variables to study hyperbolic structures on 3-manifolds derived from punctured surfaces.

Findings

Cluster y-variables solve edge-gluing conditions

Application to hyperbolic structures on mapping tori

Insights into the completeness of hyperbolic structures

Abstract

We advocate the use of cluster algebras and their y-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster y-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.