# Rigorous computations with an approximate Dirichlet domain

## TL;DR

This paper demonstrates that approximate Dirichlet domains can be effectively used in hyperbolic geometry computations, explaining the success of software like SnapPea and improving algorithms for hyperbolic 3-manifolds.

## Contribution

It establishes conditions under which approximate Dirichlet domains are as effective as exact ones and enhances algorithms for the length spectrum of hyperbolic 3-manifolds.

## Key findings

- Approximate Dirichlet domains can replace exact ones under certain assumptions.
- The approach explains the empirical success of SnapPea.
- Improved algorithm for constructing the length spectrum of hyperbolic 3-manifolds.

## Abstract

In this paper we address some problems concerning an approximate Dirichlet domain. We show that under some assumptions the 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 im- plemented 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 an improvement in the algorithm for rigorous construction of the length spectrum of a hyperbolic 3-manifold.