On volumes of hyperbolic orbifolds

Abstract

We use H C Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group to construct an explicit lower bound for the volume of a hyperbolic n-orbifold. 57M50, 57N16; 20H10, 22E40 IntroductionLet H n denote hyperbolic n-space, the unique simply connected n-dimensional Riemannian manifold of constant sectional curvature 1. A hyperbolic n-orbifold Q is a quotient H n = , where represents a discrete group of orientation-preserving isometries. A hyperbolic n-orbifold is a manifold when contain no elements of finite order. Martin [17] constructed a lower bound for r n , the largest number such that every hyperbolic n-manifold contains a round ball of that radius; see also Friedland and Hersonsky [8]. From this one can compute, in each dimension, an explicit lower bound for the volume of a hyperbolic n-manifold.The purpose of this paper is to give an explicit lower bound for the volume of a hyperbolic n-orbifold, again depending only on dimension. The result of this article is more general than what was achieved in the prequel [1]. Our work also significantly improves upon the volume bounds of [1; 17], even though we consider a larger category of orbit spaces.We define a Riemannian submersion W SO o .n; 1/= ! H n = , where the connected component SO o .n; 1/ of the identity in the Lie group O.n; 1/ is isomorphic to the full group of orientation-preserving isometries of H n . The study of the volume of a hyperbolic orbifold is thereby reduced to the study of the covolume of a lattice in a Lie group.Wang [23] showed that the covolume of a lattice in a semisimple Lie group that contains no compact factor can be bounded below by the volume of ball with a radius that depends only on the group itself. We estimate the sectional curvature of SO o .n; 1/ and apply a comparison theorem due to Gunther (see eg Gallot, Hulin and Lafontaine [10]), to produce a lower bound for VolOESO o .n; 1/= . The following theorem gives our main result.

On volumes of quaternionic hyperbolic n-orbifolds

“…In Section 6, we reestimate the bound of the sectional curvatures of SO o (n, 1) and SU(n, 1). These new bounds imply slight improvements of the results in [2,3].…”

“…Recently Adeboye and Wei reconsidered the question of lower bound for the volume of a real hyperbolic orbifold with the tools of Lie group and Riemannian submersion [2]. They obtained the following theorem.…”

“…Motivated by the ideas of Adeboye and Wei in [2,3], we will consider the question of lower bound for the volume of a quaternionic hyperbolic orbifold with the tools of Lie group and Riemannian submersion.…”

“…As in [2,3], the volume bounds for hyperbolic orbifolds provide information on the order of the symmetry groups of hyperbolic manifolds. Following Hurwitz's formula for groups acting on surfaces, we have the following corollary.…”

Quasiconformal Homogeneity after Gehring and Palka

“…One may use estimates of Adeboye-Wei [1] to obtain an explicit lower bound, for all n, on the diameter of any hyperbolic n-orbifold. Since Theorem 3.1 and McMullen's Rigidity Theorem (Theorem 3.4) imply that every quasiconformal automorphism of a uniformly quasiconformally homogeneous hyperbolic manifold is homotopic to an isometry, one should be able to follow the proof of Theorem 4.4 to produce a lower bound on K n .…”