Hyperbolic four-manifolds, colourings and mutations

TL;DR

This paper introduces a novel approach to understanding hyperbolic 4-manifolds through orbifold covers and facet colourings, enabling the construction of manifolds with specific cusp types and minimal volumes.

Contribution

It develops a method to view hyperbolic 4-manifolds as orbifold covers of Coxeter polytopes with facet colourings and describes mutations along geodesic sub-manifolds, leading to new minimal volume examples.

Findings

Constructed hyperbolic 4-manifolds with single non-toric and toric cusps.

Produced manifolds with twice the minimal volume among all such manifolds.

Developed a framework for mutations along geodesic sub-manifolds.

Abstract

We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing the result of mutations along $\mathcal{N}$. As an application of our method, we construct an example of a complete orientable hyperbolic $4$-manifold $\mathcal{X}$ with a single non-toric cusp and a complete orientable hyperbolic $4$-manifold $\mathcal{Y}$ with a single toric cusp. Both $\mathcal{X}$ and $\mathcal{Y}$ have twice the minimal volume among all complete orientable hyperbolic $4$-manifolds.