On invertible generating pairs of fundamental groups of graph manifolds

TL;DR

This paper investigates the conditions under which the fundamental groups of graph manifolds have invertible generating pairs, linking this property to the Heegaard genus of the manifold.

Contribution

It establishes a precise characterization: a graph manifold has Heegaard genus 2 if and only if its fundamental group admits an invertible generating pair.

Findings

Graph manifolds of Heegaard genus 2 have invertible generating pairs.

Invertible generating pairs correspond to automorphisms of the fundamental group.

Characterization of Heegaard genus 2 in terms of group automorphisms.

Abstract

We study invertible generating pairs of fundamental groups of graph manifolds, that is, pairs of elements (g,h) for which the map g --> g^{-1}, h --> h^{-1} extends to an automorphism. We show in particular that a graph manifold is of Heegaard genus 2 if and only if its fundamental group has an invertible generating pair.