HYPERBOLIC FOUR-MANIFOLDS WITH ONE CUSP

We introduce an 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 a (c) 1/2 V grows like C (V ln V) for any fixed k. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.