Unions of 3-punctured spheres in hyperbolic 3-manifolds

TL;DR

This paper classifies the topological configurations of unions of 3-punctured spheres in hyperbolic 3-manifolds, revealing their uniqueness and bounds on cusp moduli, advancing understanding of hyperbolic manifold structures.

Contribution

It provides a comprehensive classification of unions of 3-punctured spheres and analyzes their occurrence and geometric bounds in hyperbolic 3-manifolds.

Findings

Classified topological types of unions of 3-punctured spheres

Identified unique occurrences in specific hyperbolic 3-manifolds

Established bounds on cusp moduli for linearly placed spheres

Abstract

We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions appears only in a single hyperbolic 3-manifold or Dehn fillings of a single hyperbolic 3-manifold. Furthermore, we investigate bounds of the moduli of adjacent cusps for the union of linearly placed 3-punctured spheres.