On the doubled tetrus

TL;DR

This paper constructs a sixfold cover of the double of the tetrus, a hyperbolic 3-manifold, demonstrating its fibering over the circle and exploring arithmetic properties of related doubles and twisted doubles.

Contribution

It introduces a specific sixfold cover of the double of the tetrus and analyzes its fibering and arithmeticity, advancing understanding of hyperbolic 3-manifold covers.

Findings

The sixfold cover fibers over the circle with genus 19 surface

Doubles and twisted doubles of tripus and tetrus are arithmetic

Identifies consequences for families of covers of these manifolds

Abstract

The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic 3-manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with fiber a closed surface of genus 19. We also record arithmeticity of the doubles and certain twisted doubles of the tripus and tetrus, and point out some consequences regarding families of covers.