Generating pairs of 2-bridge knot groups

TL;DR

This paper explores the Nielsen equivalence classes of generating pairs in Kleinian groups, revealing infinite classes in hyperbolic 2-bridge knot groups and characterizing equivalences in HNN-extensions.

Contribution

It establishes the existence of infinitely many Nielsen classes in hyperbolic 2-bridge knot groups and characterizes Nielsen equivalences in HNN-extensions.

Findings

Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes.

Existence of closed hyperbolic 3-manifolds with N Nielsen classes.

Nielsen equivalence in HNN-extensions is characterized by obvious reasons.

Abstract

We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural number N there is a closed hyperbolic 3-manifold whose fundamental group has N distinct Nielsen classes of generating pairs. - Two pairs of elements of a fundamental group of an HNN-extension are Nielsen equivalent iff they are so for the obvious reasons.