The Space of Hyperbolic Manifolds and the Volume Function

Abstract

In the whole of this chapter we shall always suppose manifolds are connected and oriented. It follows from the Gauss-Bonnet formula B.3.3 (for n = 2) and from the Gromov-Thurston theorem C.4.2 (for n ≥ 3) that the volume of a hyperbolic manifold is a topological invariant. Moreover B.3.3 implies that such an invariant is (topologically) complete for n = 2 in the compact case, and it may be proved that in the finite-volume case it becomes complete together with the number of cusp ends (“punctures”). Hence the problem of studying the volume function arises quite naturally: this is the aim of the present chapter.