Rigorous computations with an approximate Dirichlet domain
If a topological 3-manifold is known to be hyperbolic (for example if solutions to Thurston's gluing equations are known) then the usual way to compute geometric invariants which cannot be immediately deduced from the existence proof such as length spectrum goes through computing first a Dirichlet domain for the action on hyperbolic 3-space (this is implemented in particular in SnapPea). On the other hand it is hard to compute a certifiable Dirichlet domain for manifolds with very thin parts. The present paper shows that for certain problems (in particular computing the length spectrum) it is sufficient to stop when one has computed up to an upper bound on the volume of the remaining facets, justifying rigorously the efficiency shown in practice by SnapPea's algorithm. More precisely it shows that the graph obtained by taking all group elements which move a well-chosen basepoint at distance at most \(R\) and connecting those for which the translates of the approximate Dirichlet domain \(D'\) is connected; thus, to compute the elements which will contribute to the length spectrum up to length \(R\) it is sufficient to use this graph.
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