Volumes of hyperbolic orbifolds.

Abstract

Various authors have previously established explicit lower bounds on the volume of hyperbolic orbifolds of dimension three and of hyperbolic manifolds of any dimension. A hyperbolic <italic>n</italic>-orbifold is a quotient of hyperbolic <italic>n</italic>-space by a discrete group Gamma of isometries of <blkbd>Hn</blkbd> . In pursuit of volume bounds for hyperbolic orbifolds, the presence of torsion elements in the group Gamma raise challenges that cannot be handled purely by the techniques developed in the manifold case. We will establish lower bounds on the volume of hyperbolic orbifolds that depend on dimension and the maximum order of the elliptic elements of Gamma. We also discuss progress in refining this result as well as applications.