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 $C^{VlogV}$ for any fixed $k$. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.

24 pages, 15 figures, typos corrected; Geom. and Funct. Anal., 2013

References in corpus (4)

• Simplicial volume and fillings of hyperbolic manifolds
• Pinched smooth hyperbolization
• On the optimality of the ideal right-angled 24-cell
• On the number of ends of rank one locally symmetric spaces

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