Geometrical finiteness with variable negative curvature.

*(English)*Zbl 0877.57018The paper under review deals with discrete subgroups of isometry groups of pinched Hadamard manifolds, that is complete, simply connected Riemannian manifolds all of whose sectional curvatures lie between two negative constants. The main subject of the paper is the notion of “geometrical finiteness” for such groups. In the case of the hyperbolic space \(\mathbb{H}^n\) of dimension \(n=3\) five equivalent definitions of geometrical finiteness are known since works of L. Ahlfors, A. Beardon and B. Maskit, A. Marden, and W. Thurston. The case of higher dimension (\(n>3\)) was completely described by B. H. Bowditch’s previous paper [J. Funct. Anal. 113, No. 2, 245-317 (1993; Zbl 0789.57007)].

Historically the term “geometrically finite” was given by L. Ahlfors for those groups which possess a finitely-sided fundamental Dirichlet polyhedron in the hyperbolic space \(\mathbb{H}^3\). However, there is no evidence to introduce an analogous definition in the case of variable curvature because of some known examples due to W. Goldman and J. Parker. All other four definitions turn out to work and it is proven in the paper that they all are equivalent. Below we outline briefly these definitions.

F1. \(\Gamma\) is geometrically finite if the orbifold with boundary \(M_C(\Gamma)= X\cup\Omega/\Gamma\) is compact, where \(\Omega\) is the discontinuity domain for the action of \(\Gamma\) on the ideal boundary of \(X\).

F2. \(\Gamma\) is said to be geometrically finite (“à la Beardon-Maskit”) if the limit set \(\Lambda\) of \(\Gamma\) consists entirely of approximation (conical) limit points and bounded parabolic fixed points.

F3 used to be a condition on the finiteness of Dirichlet polyhedrons and remains unknown.

F4. The “thick part” of the “convex core” of \(X/\Gamma\) is compact.

F5. For some \(\eta>0\), the uniform \(\eta\)-neighbourhood of the convex core has finite volume, and there is a bound on the orders of finite subgroups of \(\Gamma\) (the latter turned out to be an essential condition even in the case of constant curvature in higher dimension).

Historically the term “geometrically finite” was given by L. Ahlfors for those groups which possess a finitely-sided fundamental Dirichlet polyhedron in the hyperbolic space \(\mathbb{H}^3\). However, there is no evidence to introduce an analogous definition in the case of variable curvature because of some known examples due to W. Goldman and J. Parker. All other four definitions turn out to work and it is proven in the paper that they all are equivalent. Below we outline briefly these definitions.

F1. \(\Gamma\) is geometrically finite if the orbifold with boundary \(M_C(\Gamma)= X\cup\Omega/\Gamma\) is compact, where \(\Omega\) is the discontinuity domain for the action of \(\Gamma\) on the ideal boundary of \(X\).

F2. \(\Gamma\) is said to be geometrically finite (“à la Beardon-Maskit”) if the limit set \(\Lambda\) of \(\Gamma\) consists entirely of approximation (conical) limit points and bounded parabolic fixed points.

F3 used to be a condition on the finiteness of Dirichlet polyhedrons and remains unknown.

F4. The “thick part” of the “convex core” of \(X/\Gamma\) is compact.

F5. For some \(\eta>0\), the uniform \(\eta\)-neighbourhood of the convex core has finite volume, and there is a bound on the orders of finite subgroups of \(\Gamma\) (the latter turned out to be an essential condition even in the case of constant curvature in higher dimension).

Reviewer: L.Potyagailo (Villeneuve d’Ascq)

##### MSC:

57S25 | Groups acting on specific manifolds |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

53C20 | Global Riemannian geometry, including pinching |

57R99 | Differential topology |

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\textit{B. H. Bowditch}, Duke Math. J. 77, No. 1, 229--274 (1995; Zbl 0877.57018)

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##### References:

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