On quotient orbifolds of hyperbolic 3-manifolds of genus two

TL;DR

This paper classifies quotient orbifolds of genus two hyperbolic 3-manifolds, revealing their topological types and singular sets, and constructs examples of manifolds as branched covers of different knots.

Contribution

It provides a detailed classification of quotient orbifolds of genus two hyperbolic 3-manifolds and constructs examples of manifolds as branched covers of distinct knots.

Findings

Quotient orbifolds are either 3-sphere, lens space, or prism manifold.

The singular set structure is described for each isometry group.

An infinite family of manifolds as branched covers of different knots is constructed.

Abstract

We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the hyperbolic 2-fold branched coverings of 3-bridge links. If the 3-bridge link is a knot, we prove that the underlying topological space of the quotient orbifold is either the 3-sphere or a lens space and we describe the combinatorial setting of the singular set for each possible isometry group. In the case of 3-bridge links with two or three components, the situation is more complicated and we show that the underlying topological space is the 3-sphere, a lens space or a prism manifold. Finally we present a infinite family of hyperbolic 3-manifolds that are simultaneously the 2-fold branched covering of two inequivalent knot, one with bridge number three and the other one with bridge number strictly greater than three.