Abstract
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 C V ln V for any fixed k. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.
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A. Kolpakov was supported by the SNSF researcher scholarship PBFRP2-145885 and the SNSF project “Discrete hyperbolic geometry” no. 200020-144438/1. B. Martelli was supported by the Italian FIRB project “Geometry and topology of low-dimensional manifolds”, RBFR10GHHH.
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Kolpakov, A., Martelli, B. Hyperbolic four-manifolds with one cusp. Geom. Funct. Anal. 23, 1903–1933 (2013). https://doi.org/10.1007/s00039-013-0247-2
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DOI: https://doi.org/10.1007/s00039-013-0247-2