TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 4, April 1999, Pages 1507–1530 S 0002-9947(99)02113-3 LIMIT SETS OF DISCRETE GROUPS OF ISOMETRIES OF EXOTIC HYPERBOLIC SPACES KEVIN CORLETTE AND ALESSANDRA IOZZI Abstract. Let Γ be a geometrically finite discrete group of isometries of hy- perbolic space H n F , where F = R, C, H or O (in which case n = 2). We prove that the critical exponent of Γ equals the Hausdorff dimension of the limit sets Λ(Γ) and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven. 1. Introduction In a previous paper [C], the first author studied limit sets of discrete groups of isometries of rank one symmetric spaces of noncompact type. These symmet- ric spaces are the hyperbolic spaces associated with the reals, complexes, quater- nions, or Cayley numbers. The main point of interest in that paper was the Haus- dorff dimension (in an appropriate sense) of the limit set and its relationships with other invariants, such as the smallest eigenvalue of the Laplacian acting on square- integrable functions and the exponent of growth of the group. In the cases where the group of all isometries of the symmetric space has Kazhdan’s property, this led to conclusions about limitations on the possible exponents of growth, among other things. The most precise results were obtained for a class of groups which, in that paper, were called geometrically cocompact; these groups are also known as convex cocompact groups, or geometrically finite groups without cusps. In particular, the exponent of growth was shown to be equal to the Hausdorff dimension of the limit set, from which one could draw conclusions about the topological structure of the hyperbolic manifolds corresponding to such groups. In this paper, we will study the case of arbitrary geometrically finite groups of isometries of these symmetric spaces. The main result is that the exponent of growth coincides with the Hausdorff dimension of the limit set; as a conse- quence, when these are sufficiently large, the smallest eigenvalue of the Laplacian is a quadratic function of either of them and can be shown to correspond to a square-integrable eigenfunction. (This latter fact will be exploited in a forthcoming paper of the first author to give estimates on the topological codimension of the Received by the editors February 27, 1995 and, in revised form, April 15, 1997. 1991 Mathematics Subject Classification. Primary 58F11; Secondary 53C35, 58F17. K. C. received support from a Sloan Foundation Fellowship, an NSF Presidential Young In- vestigator award, and NSF grant DMS-9203765. A. I. received support from NSF grants DMS 9001959, DMS 9100383 and DMS 8505550. c 1999 American Mathematical Society 1507 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use