Hyperbolic four-manifolds with one cusp

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.