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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
A new invariant on hyperbolic Dehn surgery space

James G Dowty

Algebraic & Geometric Topology 2 (2002) 465–497

arXiv: math.GT/0207060

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
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57M27
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Publication
Received: 24 October 2001
Revised: 24 May 2002
Accepted: 6 June 2002
Published: 22 June 2002
Authors
James G Dowty
Department of Mathematics
University of Melbourne
Parkville, 3052
Australia