# Definition of cusped manifold?

There is much talk about hyperbolic cusped 3-manifolds, but almost no definition of what a cusped manifold is.

One definition I found was that it is a result of a parabolic transformation on H^n, fixing the infinity point (?).

Is there a more intuitive definition for the 3-dimensional case. What about a general n-dimensional case. Is there a definition of a cusp?

## 1 Answer

Cusped manifolds are noncompact complete hyperbolic manifolds with finite Riemannian volume.

More precisely, a cusped hyperbolic n-manifold is a Riemannian manifold (without boundary) of constant negative curvature, which is metrically complete and has finite Riemannian volume, but is not compact. If you prefer to look at group actions, then you can see it as the quotient of n-hyperbolic space by a group of isometries acting properly discontinuously, freely, and with a fundamental domain of finite volume but no compact fundamental domain.

This definition does not depend on the dimension. The name "cusped" comes from the fact that these manifolds have the following structure: say $M$ is one such, then it retracts onto a compact submanifold $M'$ which has a boundary consisting of flat manifolds $T_1, \ldots, T_h$. The rest of the manifold consists of so-called "cusps", which are warped products $T_i \times [1, +\infty[$ with the metric $(dx^2 + dt^2)/t^2$ (where $dx$ is a flat metric on $T_i$). The uniformisation of a cusp is as follows: take the upper-half plane model for hyperbolic space, so that it decomposes as $\mathbb H^n = E^{n-1} \times ]0, +\infty[$ where $E^{n-1}$ is (n-1)-Euclidean space. Let $\Lambda$ be a $n-1$-dimensional torsion-free crystallographic group (a subgroup of Euclidean isometries of $E^{n-1}$ with a compact fundamental tile, for example $\mathbb Z^{n-1}$ acting by translations). It acts (by parabolic isometries) on $\mathbb H^n$ preserving the subsets $E^{n-1}\times\{t\}$; then the quotient $\Lambda \backslash E^{n-1} \times[1,+\infty[$ is a cusp.

In 3 dimensions, by Perelman's solution the Thurston geometrisation conjecture there is a topological characterisation of hyperbolic cusped manifolds (I am writing from memory here, there might be imprecisions): they are exactly the interior of irreducible compact manifolds with a nonempty boundary consisting of tori, which contain no other essential tori up to homotopy, and no embedded annuli connecting two essential curves on distinct boundary components (In this setting irreducible means that every sphere bounds a ball and every disc bounds a half-ball). (EDIT: added the condition on annuli and precised that only the interior is hyperbolic above)

• Thank you for the great reply. I am interested in the topological definition. Do you perhaps know a reference where I can look it up? – Jake B. Nov 21 '16 at 18:40
• Also: in line 1 you claim cusped manifolds are noncompact and at the end (line-3) you claim cusped manifolds are compact. – Jake B. Nov 21 '16 at 19:40
• I (hopefully) clarified the compact/non compact thing in 3D. – Jean Raimbault Nov 22 '16 at 7:45
• As for a reference for the Geometrisation theorem, you can go look at Bruno Martelli's book at dm.unipi.it/~martelli/Geometric_topology.pdf, especially DEfn. 9.4.17 and Corollary 12.9.6 (there is no proof of Geometrisation there, but it does give furthere references and anyway I think the proof is besides the point in this case) – Jean Raimbault Nov 22 '16 at 7:48
• Perelman's work is not relevant to the topological characterization of hyperbolic cusped manifolds. Thurston already proved this characterization. – Lee Mosher Nov 22 '16 at 15:05