On volumes of complex hyperbolic orbifolds

Abstract

We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.

On volumes of quaternionic hyperbolic n-orbifolds

“…Theorem 1.2. (Theorem 0.1 of Adeboye and Wei [3]) The volume of a complex hyperbolic n -orbifold is bounded below by C(n), an explicit constant depending only on dimension, given by C(n) = 2 n 2 +n+1 π n 2 (n − 1)! (n − 2)!…”

“…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.…”

“…Section 4 aims to obtain the bound of the sectional curvatures of Sp(n, 1) with respect to the scaled canonical metric. In order to obtain better estimate, we will use some formulae of the connection and curvature which are slight different from those in [3]. Section 5 contains the proof of Theorem 1.3 with the similar route map employed in [3].…”

“…In order to obtain better estimate, we will use some formulae of the connection and curvature which are slight different from those in [3]. Section 5 contains the proof of Theorem 1.3 with the similar route map employed in [3]. In Section 6, we reestimate the bound of the sectional curvatures of SO o (n, 1) and SU(n, 1).…”