Hyperbolic four-manifolds, colourings and mutations

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.

24 pages, 11 figures; classification in Proposition 3.2 is incomplete: see arXiv:2009.09995 for correction

References in corpus (3)

• Some hyperbolic three-manifolds that bound geometrically
• Symmetries of hyperbolic 4-manifolds
• A four dimensional hyperbolic link complement in a standard $S^4$

Cited by in corpus (5)

• Cusps of hyperbolic 4-manifolds and rational homology spheres
• Hyperbolic 4-manifolds with perfect circle-valued Morse functions
• Infinitesimal Rigidity for Cubulated Manifolds
• Cusps and Commensurability Classes of Hyperbolic 4-Manifolds
• Some cusp-transitive hyperbolic 4-manifolds