Hyperbolic three-manifolds that embed geodesically

TL;DR

The paper proves that certain hyperbolic 3-manifolds tessellated into right-angled polyhedra can be embedded geodesically into hyperbolic 4-manifolds tessellated into higher-dimensional right-angled polytopes, with volume bounds.

Contribution

It establishes the existence of geodesic embeddings of specific hyperbolic 3-manifolds into hyperbolic 4-manifolds with explicit volume bounds and tessellation properties.

Findings

Every such 3-manifold embeds into a 4-manifold tessellated by 120-cells and 24-cells.

Volume of the 4-manifold is less than 2^49 times the volume of the 3-manifold.

The Borromean link complement bounds a hyperbolic 4-manifold geometrically.

Abstract

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.