Hyperbolic four-manifolds with one cusp

TL;DR

This paper presents an algorithm to construct hyperbolic four-manifolds from cubulations, resulting in the first examples of such manifolds with a single cusp and analyzing their growth in number relative to volume.

Contribution

It introduces a novel algorithm linking cubulations to hyperbolic four-manifolds and constructs the first known examples with one cusp.

Findings

Constructed the first hyperbolic four-manifolds with one cusp.

Showed the number of k-cusped hyperbolic four-manifolds grows like C^{V log V}.

Proved the 3-torus bounds a hyperbolic manifold geometrically.

Abstract

We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of $k$-cusped hyperbolic four-manifolds with volume smaller than V grows like $C^{V log V}$ for any fixed $k$. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.