Hyperbolic four-manifolds with one cusp

By A. Kolpakov, Bruno Martelli. Geometric and Functional Analysis, 2013

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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 ⩽ V grows like CV ln V for any fixed k. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/259d225b0d4af8b49180b86fbdec4f8f5b21aa0b