Rigorous computations with an approximate Dirichlet domain

Abstract

In this paper we address some problems concerning an approximate Dirichlet domain. We show that under some assumptions an approximate Dirichlet domain can work equally well as an exact Dirichlet domain. In particular, we consider a problem of tiling a hyperbolic ball with copies of the Dirichlet domain. This problem arises in the construction of the length spectrum algorithm which is implemented by the computer program SnapPea. Our result explains the empirical fact that the program works surprisingly well despite it does not use exact data. Also we demonstrate a rigorous verification whether two words of the fundamental group of a hyperbolic 3-manifold are the same or not.