On Finiteness of Kleinian Groups in General Dimension

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper of Bishop and Jones to show that conformal finiteness implies geometric finiteness unless the set of limit points is of Hausdorff dimension n. Furthermore we show that, for a given Kleinian group Γ, conformal finiteness is equivalent to the existence of a metric of finite geometry on the Kleinian manifold Ω(Γ)/Γ. 1991 Mathematics Subject Classification: Primary 53A30; Secondary 30F40, 58J60, 53C21. §0. Introduction A discrete subgroup Γ of the group of conformal transformations of the unit sphere S is called a Kleinian group if there is a non-empty domain Ω(Γ) of discontinuity in S. The group Γ also acts as a subgroup of the group of hyperbolic isometries of the unit ball B. Assuming the group has no torsion elements, the quotient is a hyperbolic manifold B/Γ, which is bounded at infinity by a Kleinian manifold Ω(Γ)/Γ which is a manifold with a locally conformally flat structure. There is a useful notion of geometric finiteness for Kleinian groups that assures nice properties of the geometric quotient. There are several equivalent forms of this condition. According to one formulation, a Kleinian group Γ is said to be geometrically finite if the limit set Λ(Γ) consists only of conical limit points and cusped limit points. It is a natural problem to find useful criteria to assure geometric finiteness. In dimension two, a Kleinian group is said to be analytically finite if the Riemann surface Ω(Γ)/Γ is of finite type (that is to say a union of a finite number of closed Riemann surfaces each with finite number of punctures). Recently Bishop and Jones ([5]) showed that, for an analytically finite group Γ, Λ(Γ) has Hausdorff Research is supported in part by NSF Grant DMS-0070542 and a Guggenheim Foundation Fellowship. Research is supported in part by NSF Grant DMS-9803399 and Slaon Fellowships BR-3818. Research is supported in part by NSF Grant DMS-0070526. Typeset by AMS-TEX 1 2 ON FINITENESS OF KLEINIAN GROUPS IN GENERAL DIMENSION dimension less than two if and only if Γ is geometrically finite. An important part of their work is the construction of an invariant Lipschitz graph which serves to relate the geometry of the hyperbolic manifold B/Γ to the geometry of the Riemann surface Ω(Γ)/Γ. Recently we studied locally conformally flat 4-manifolds and obtained some finiteness for certain class of such manifolds [7] [8]. In those works [8] the holonomy representation of the fundamental group of such manifolds as Kleinian group played a key role in our understanding of the structure of such manifolds. In dimension higher than two, Kleinian groups have been studied mostly in conjunction with hyperbolic structure. Our motivation is to investigate the close relation between the geometry of the hyperbolic manifolds B/Γ and the geometry of the Kleinian manifold Ω(Γ)/Γ for a given Kleinian group Γ. For our purpose, we introduce a notion of conformal finiteness (see Definition 3.2 in Section 3), which is the natural analogue of the notion of analytic finiteness to higher dimensions. Then we extend the theorem of Bishop and Jones to Kleinian groups in higher dimension. Theorem 0.1. Suppose that Γ is a nonelementary, conformally finite Kleinian group on S, then Γ is geometrically finite if and only if the limit set of Γ has Hausdorff dimension strictly smaller than n. Recall the celebrated finiteness theorem of Ahlfors [2] and Bers [3] [4], which states that a finitely generated Kleinian group in dimension two is analytically finite. This finiteness theorem fails to hold in higher dimensions as pointed out by the examples of Kapovich [11], and Kapovich and Potyagailo [12]. On the other hand, a result of Jarvi and Vuorinen [10] shows that in general dimensions, the limit set of a finitely generated Kleinian group is uniformly perfect. The latter condition is equivalent, in dimension two, to the condition of analytic finiteness of the group. Moreover in [1] Ahlfors showed some weak finiteness for Kleinian groups in higher dimension: if Γ is finitely generated, then the dimension of the space of certain class of mixed tensor densities, automorphic under Γ, is finite (see also [14] of Hiromi Ohtake). Therefore, it is interesting to search for the appropriate version of the finiteness result in higher dimension, particularly for those Kleinian groups with small limit sets. We take a first step in that direction by characterizing the conformally finite ends by geometric conditions. As another application of the Lipschitz graph construction, we have Theorem 0.2. Given a Kleinian group Γ, the Kleinian manifold Ω(Γ)/Γ is of finite geometry for some metric in the conformal class if and only if Γ is conformally finite. By finite geometry here we mean that its curvature and covariant derivatives of curvature is bounded, and its volume is finite. We would like to thank Francis Bonahon and Feng Luo for informative discussions and interest in this work. The second author would like to thank MSRI for the hospitality. This note is completed when the second author is visiting MSRI. §1. Construction of the Lipschitz Graph CHANG, QING AND YANG 3 The following construction, by completely elementary means, of the invariant Lipschitz graph over a domain of discontinuity Ω(Γ) of a Kleinian group Γ is based on the idea of Bishop and Jones’ in [5]. Take a small positive number ǫ0 and consider a collection of balls {Bα} such that (1.1) Bα = B(xα, dα) and dα = ǫ0 · dist(xα, L(Γ)) for each point xα ∈ Ω(Γ), where diameters and distances are all measured on S with the standard metric g0. To construct an invariant graph we would enlarge the collection to take in all images of Bα under the group Γ and denote the collection of balls by B(Γ). Set (1.2) G(Γ) = ∂( ⋃