Hyperbolic four-manifolds with one cusp

Alexander Kolpakov (1) , Bruno Martelli (2)

1 Department of Mathematics, Vanderbilt University
2 Dipartimento di Matematica [Pisa]

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.

Mots clés: 4-manifold; hyperbolic manifold; geometric boundary; cusp section

Domaines: Géométrie métrique [math.MG]

Alexander Kolpakov, Bruno Martelli. Hyperbolic four-manifolds with one cusp. Geometric And Functional Analysis, 2013, On-line first, pp.1-33. ⟨10.1007/s00039-013-0247-2⟩. ⟨hal-00837649v2⟩