##
**Geometrical finiteness with variable negative curvature.**
*(English)*
Zbl 0877.57018

The 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 |

### Citations:

Zbl 0789.57007
PDF
BibTeX
XML
Cite

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

Full Text:
DOI

### References:

[1] | L. V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups , Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 251-254. JSTOR: · Zbl 0132.30801 |

[2] | M. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature , J. Differential Geom. 18 (1983), no. 4, 701-721. · Zbl 0541.53036 |

[3] | W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of Nonpositive Curvature , Progr. Math., vol. 61, Birkhäuser, Boston, 1985. · Zbl 0591.53001 |

[4] | A. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra , Acta Math. 132 (1974), 1-12. · Zbl 0277.30017 |

[5] | B. H. Bowditch, Geometrical finiteness for hyperbolic groups , J. Funct. Anal. 113 (1993), no. 2, 245-317. · Zbl 0789.57007 |

[6] | B. H. Bowditch, Discrete parabolic groups , J. Differential Geom. 38 (1993), no. 3, 559-583. · Zbl 0793.53029 |

[7] | B. H. Bowditch, Some results on the geometry of convex hulls in manifolds of pinched negative curvature , Comment. Math. Helv. 69 (1994), no. 1, 49-81. · Zbl 0967.53022 |

[8] | M. R. Bridson, Geodesics and curvature in metric simplicial complexes , Group Theory from a Geometrical Viewpoint (Trieste, 1990) eds. E. Ghys, A. Haefliger, and A. Verjovsky, World Scientific, River Edge, NJ, 1991, pp. 373-463. · Zbl 0844.53034 |

[9] | J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry , North-Holland Math. Lib., vol. 9, North-Holland, Amsterdam, 1975. · Zbl 0309.53035 |

[10] | D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces , Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, 1984) ed. D. B. A. Epstein, London Math. Soc. Lecture Note Series, vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113-253. · Zbl 0612.57010 |

[11] | W. M. Goldman, Complex hyperbolic Kleinian groups , Complex Geometry (Osaka, 1990), Lecture Notes in Pure and Appl. Math., vol. 143, Dekker, New York, 1993, pp. 31-52. · Zbl 0793.53069 |

[12] | W. M. Goldman and J. R. Parker, On the horospherical geometry of complex hyperbolic space , preprint, Univ. of Maryland, 1991. |

[13] | E. Heintze and H. C. Im Hof, Geometry of horospheres , J. Differential Geom. 12 (1977), no. 4, 481-491. · Zbl 0434.53038 |

[14] | A. Marden, The geometry of finitely generated kleinian groups , Ann. of Math. (2) 99 (1974), 383-462. JSTOR: · Zbl 0282.30014 |

[15] | J. R. Parker, Dirichlet polyhedra for parabolic cyclic groups acting on complex hyperbolic space , preprint, Warwick Univ., 1992. · Zbl 0762.51009 |

[16] | M. B. Phillips, Dirichlet polyhedra for cyclic groups in complex hyperbolic space , Proc. Amer. Math. Soc. 115 (1992), no. 1, 221-228. JSTOR: · Zbl 0768.53033 |

[17] | M. Spivak, A Comprehensive Introduction to Differential Geometry , 2nd ed., Publish or Perish, Wilmington, Del., 1979. · Zbl 0439.53001 |

[18] | W. P. Thurston, The geometry and topology of \(3\)-manifold , Princeton Univ., Department of Mathematics, 1979. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.