Classification of parabolic generating pairs of Kleinian groups with two parabolic generators

TL;DR

This paper provides an alternative proof to Agol's classification of parabolic generating pairs in non-free Kleinian groups with two parabolic generators, and applies it to classify epimorphisms between 2-bridge knot groups and degree one maps between hyperbolic 2-bridge link exteriors.

Contribution

It offers a new proof of a classification theorem and applies it to characterize epimorphisms and degree one maps in knot and link exteriors.

Findings

Complete classification of parabolic generating pairs in non-free Kleinian groups

Characterization of epimorphisms between 2-bridge knot groups

Characterization of degree one maps between hyperbolic 2-bridge link exteriors

Abstract

We give an alternative proof to Agol's classification of parabolic generating pairs of non-free Kleinian groups generated by two parabolic transformations. As an application, we give a complete characterisation of epimorphims between $2$-bridge knot groups and a complete characterisation of degree one maps between the exteriors of hyperbolic $2$-bridge links.