Practical bounds for a Dehn parental test

Posted as arXiv:1504.01674.

Published in Proc. Amer. Math. Soc. 147, 427--442, 2019.

Consider the following problem: given two three-manifolds \(M\) and \(N,\) to determine whether or not \(N\) is a Dehn filling of \(M.\) Ideally, if \(M\) and \(N\) were finite-volume complete orientable hyperbolic three-manifolds, one would like to apply the work of Hodgson and Kerckhoff directly. Unfortunately this would require drilling out geodesics from said manifolds. There are no known efficient procedures for doing so, given known efficient representations of geodesics (for instance, as axes of deck transformations in the universal cover \(\mathbb{H}^3.\)) In this paper I addressed this difficulty by pushing Hodgson and Kerckhoff’s bounds further, and got bounds that work when one is unable to drill out geodesics.

One novelty to me when writing this paper was using formal methods. I used the software suite Coq and its numerical analysis module Coquelicot to verify some technical bounds on integrals. Another novelty was the theory of computation on real numbers. If real numbers are just Cauchy sequences up to equivalence, then, computably, the trichotomy law does not hold. In general the only equality test you can hope to run is essentially to check all the decimal digits. If the given numbers are in fact equal, this equality test will not terminate. Instead, all you can assume is a dichotomy law: if you can demonstrate \(x < y,\) then for all \(z\) you can demonstrate \(x < z\) or \(z < y.\) (Here the “or” is taken as a “live” concurrent test, where progress is always made after some time on determining both disjuncts.) My procedure works using this model.