Algebraic and Geometric Topology 2 (2002), paper no. 23, pages 465-497.

A new invariant on hyperbolic Dehn surgery space

James G. Dowty

Abstract. In this paper we define a new invariant of the incomplete hyperbolic structures on a 1-cusped finite volume hyperbolic 3-manifold M, called the ortholength invariant. We show that away from a (possibly empty) subvariety of excluded values this invariant both locally parameterises equivalence classes of hyperbolic structures and is a complete invariant of the Dehn fillings of M which admit a hyperbolic structure. We also give an explicit formula for the ortholength invariant in terms of the traces of the holonomies of certain loops in M. Conjecturally this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery space of M.

Keywords. Hyperbolic cone-manifolds, character variety, ortholengths

AMS subject classification. Primary: 57M50. Secondary: 57M27.

DOI: 10.2140/agt.2002.2.465

E-print: arXiv:math.GT/0207060

Submitted: 24 October 2001. (Revised: 24 May 2002.) Accepted: 6 June 2002. Published: 22 June 2002.

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James G. Dowty
Department of Mathematics, University of Melbourne
Parkville, 3052, Australia

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