On volumes of hyperbolic orbifolds
Lower bounds for the radius \(r_{n}\) of the largest ball that can be embedded in any hyperbolic \(n\)-manifold (and, hence, lower bounds on the volumes of these spaces) have been known for some time [\textit{G. J. Martin}, J. Lond. Math. Soc., II. Ser. 40, No. 2, 257--264 (1989; Zbl 0709.30040)]. In this paper, the authors prove that the volume of a hyperbolic \(n\)-orbifold is bounded below by a constant (an explicit formula for which is computed) that depends only on the dimension of the orbifold. This significantly improves upon the above volume bounds for manifolds, as well as on previous work of the first author [\textit{I. Adeboye}, Pac. J. Math. 237, No. 1, 1--19 (2008; Zbl 1149.57026)]. In the cases when \(n\) equals either 2 or 3, the authors refine the estimates to produce better bounds than those given by the general formula. The proof of the result uses a bound developed by Wang [\textit{H.-C. Wang}, J. Differ. Geom. 3, 481--492 (1969; Zbl 0197.30403)] on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group.
scientific article; zbMATH DE number 6035475
scientific article