Constructions of surface bundles with rank two fundamental groups

TL;DR

This paper constructs hyperbolic 3-manifolds with rank two fundamental groups as surface bundles over the circle, explores their properties, and identifies an infinite family of genus two fibered knots with rank two knot groups.

Contribution

It introduces a new construction method for hyperbolic 3-manifolds with rank two fundamental groups and identifies a family of genus two fibered knots with this property.

Findings

All constructed manifolds are surface bundles over the circle with genus two fibers.

Some manifolds have Heegaard genus two.

An infinite family of genus two fibered knots with rank two knot groups is identified.

Abstract

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As a byproduct, we give an infinite family of fibered knots of genus two in the 3-sphere whose knot groups are of rank two.