On the absence of the Ahlfors and Sullivan theorems for Kleinian groups in higher dimensions

Abstract

M~ E. Kapovich and L. D. Potyagailo UDC 512.8t7 Introduction One of the fundamental results of the theory of discontinuous groups of fractional linear transformations acting on the complex plane C is the following finiteness theorem of L. Ahlfors [1, 2]: Let G be a finitely generated nonelementary discrete subgroup of PSL (2, C) acting freely on a reqion of discontinuity ~2(G). Then the quotient surface a(a)/G consists of a finite number of Riemann surfaces $1,..., S~ of finite hyperbolic area. In particular the groups rq ( Si) are finitely generated and the homotopy type of the surfaces Si is finite (i = 1,..., n). Subsequently D. Sullivan [3] strengthened Ahlfors' finiteness theorem by showing that any finiteIy generated discrete group G C PSL (2, C) has at most a finite number of cusps (i.e., conjugacy classes of maximal parabolic subgroups). Ahtfors [4] and Ohtake [5] attempted to develop analytic methods of studying the problem of finiteness of multidimensional Kleinian groups. However, the results obtained do not give any- information about either the topology of the quotient spaces of Kleinian groups or the number of cusps. In the present article we shall show that even a weakened version of Ahlfors' finiteness theorem fails in dimension 3 and also construct a counterexample to the analog of Suilivan's finiteness theorem in higher dimensions. Theorem 1. There exists a finitely generated torsion-free function group F C Mgb (S 3) with invariant component ft C ~2(F) such that the fundamental group rrl(f~/F ) is infinitely generated, Moreover the group F itself is infinitely defined. Theorem 2. There exists a finitely generated Kleinian group F I C Mbb (S 3) such that a) F' contains an infinite number of cusps (of rank 1), b) if F n is a con.formal extension of the group F' to S", then rank(H,~_a(ft(F')/F~,Z)) = ee. Thus the manifold ft(F")/F ~ has an infinite homotopy type. 1. Preliminary information Let M6b (R'~) _~ Isom (H n+l) be the group of conformal automorphisms of the n-dimensional sphere S ~ = R~ = R '~ U {~o}, where H ~+1 = {(xl,...,x,~,xn+l) E R ~+1 : x~+l > 0} is hyperbolic space. A subgroup G C M5b (S ~) is called Kleinian if the action of G is discontinuous at some point x C S ~, i.e., there exists a neighborhood U(x) such that g(U(x))NU(x) 5r 0 for only a finite number of elements g E G. The set of points where G acts discontinuously is called the discontinuity set ~Q(G) and its complement A(G) = S n \ a(G) the limit set of the group G. A Kleinian group G is called a function group if there exists a connected component f~ C f~(G) that is invariant with respect to G. If G acts freely on f~, then the quotient space M(G) = ft/G is an n-dimensional manifold. We shall denote by Y'(G) the isometric fundamental region for G [6] and by I(g) the isometric sphere g E MSb (Sn). In what follows we shall assume (if not otherwise specified) that all manifolds are three-dimensional and piecewise linear. Standard reductions by the theory of Kleinian groups and three-dimensional topology can be found in [2] and [6]-[8]. If S C