Hyperbolic four-manifolds with one cusp
We introduce an algorithm which transforms every four-dimensional cubulation into an orientable 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 -cusped hyperbolic four-manifolds with volume grows like for any fixed . As a corollary, we deduce that the -torus bounds geometrically a hyperbolic manifold.
By Margulis’ Lemma, a finite-volume complete hyperbolic -manifold has a finite number of ends called cusps, each of which is diffeomorphic to for a certain closed connected flat -manifold .
In dimension three, we may construct cusped hyperbolic manifolds in various ways, for instance by removing a knot or link complement from . There are essentially two different techniques to prove that a link complement is hyperbolic: by decomposing it into geodesic ideal hyperbolic polyhedra, or by checking that the manifold does not contain an immersed essential surface with and thus invoking geometrisation. The first method was used by Thurston in his notes [Th], where he constructed various hyperbolic -manifolds with an arbitrary number of cusps. The computer program SnapPy [SnapPy] may be used to check the hyperbolicity of any link with a reasonable number of crossings.
In higher dimensions, constructing hyperbolic manifolds is more complicated. Due to the absence of a geometrisation theorem of any kind, the hyperbolic structure on a smooth manifold needs to be established explicitly, and this is typically done either by arithmetic methods or by assembling geodesic polyhedra. The largest known census of cusped hyperbolic -manifolds is the list produced by J. Ratcliffe and S. Tschantz [RT] which contains distinct manifolds, all obtained by pairing isometrically the faces of the ideal hyperbolic -cell: these manifolds have either or cusps.
We construct here the first example of a finite-volume hyperbolic four-manifold having only one cusp. One of the motivations for this work is a result by D. Long and A. Reid [LR] which shows that, amongst the six diffeomorphism types of orientable flat -manifolds, at least two of them cannot be cusp sections of a single-cusped four-manifold (but they are sections in some multi-cusped one [N]). The authors then asked [LR, LRall] whether any single-cusped hyperbolic manifold exists in dimension . The techniques introduced in the present paper answer this question in the affirmative if the dimension is , but are not applicable in higher dimensions. We note that by a recent result of M. Stover [Stover] there are no single-cusped hyperbolic arithmetic orbifolds in dimension .
In the present paper, we show that there are plenty of single-cusped hyperbolic four-manifolds, and more generally of hyperbolic four-manifolds with any given number of cusps. Let be the number of pairwise non-homeomorphic orientable hyperbolic four-manifolds with cusps and volume at most . The main result is the following.
For every integer there are two constants , such that for all .
Let be the total number of hyperbolic four-manifolds with volume at most : it was proved in [BGLM] that for some constants .
Following P. Ontaneda [O], we say that a flat manifold bounds geometrically a hyperbolic manifold if it is diffeomorphic to a cusp section of some single-cusped hyperbolic manifold. By analysing the cusp shapes we deduce the following corollary.
The -torus bounds geometrically a hyperbolic manifold.
In fact, Ontaneda has proved that every flat manifold bounds geometrically a negatively pinched Riemannian manifold [O], but Long and Reid showed that at least two among the six orientable flat -manifolds cannot bound a hyperbolic manifold [LR]. As we said above, the -torus is the first example of a connected flat manifold of dimension that bounds a hyperbolic manifold.
The proof of Theorem 0.1 is constructive and proceeds as follows. The ideal hyperbolic -cell is a well-known ideal right-angled four-dimensional hyperbolic polytope with facets and ideal vertices: each facet is a regular ideal octahedron. The facets are naturally divided into three sets of facets each, which we colour correspondingly in green, red and blue. We produce four identical copies and of and identify the corresponding red and blue facets as described by the pattern below: