Hyperbolic Volume of Manifolds with Geodesic Boundary and Orthospectra

In this paper we describe a function Fn : R+ → R+ such that for any hyperbolic n-manifold M with totally geodesic boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\partial M \neq \emptyset}$$\end{document} , the volume of M is equal to the sum of the values of Fn on the orthospectrum of M. We derive an integral formula for Fn in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally geodesic boundary in terms of the area of the boundary.