Coordinates of the representation space in the semisimple Lie group of rank one

Abstract

In this paper we show that the space of irreducible representations from a finitely presented group into the group of isometries of a rank one symmetric space of noncompact type, embeds into R-n for some n, where the coordinates are the translation lengths of isometries in the representation. The ingredients of the proof consist of the two facts that the representation is determined by its marked length spectrum and that the nested sequence of algebraic subvarieties is stabilised at a finite step by the Noetherian property of the polynomial ring. As a minor application, we use this fact to simplify McMullen's proof about the exponential algebraic convergence of Thurston's double limit to the geometrically infinite manifold in the space of discrete faithful representations of pi(1)(S) in Iso(+) (H-R(3)).