Ortholengths and hyperbolic Dehn surgery

© 2000 Dr. James Geoffrey DowtyThis thesis studies the hyperbolic Dehn surgery space H(M) of incomplete, finite-volume hyperbolic structures on an orientable 3-manifold M which admits a complete 1-cusped hyperbolic structure. We define an ‘ortholength’ invariant on H(M) which takes values in a complex affine algebraic variety P. This invariant is easily computable and it locally parameterises H(M). For (topological) Dehn fillings of M, the ortholength invariant is a complete invariant and it determines the tube radii about the core geodesics of all but a finite number of Dehn fillings. We give a closed expression for these tube radii in terms of this invariant. Underlying this work is a geometrical construction based on the possibly-singular core which arises from generalised hyperbolic Dehn surgery on M .Given a hyperbolic structure there is a natural spine for M whose complement is the interior of a ‘tube domain’ D analogous to a Dirichlet domain based on the singular core. This domain D is a hyperbolic solid torus and the hyperbolic structure on M can be recovered from D by identifying points of its boundary. Conversely, we can try to reconstruct a hyperbolic structure on M from its ortholength invariant. To each point of P we give a representation π1(M) \to PSL2(C). We then describe a method for constructing a corresponding hyperbolic structure on M via its tube domain. This construction is quite general and it succeeds in many cases where traditional methods fail