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Abstract
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In this paper we define a new invariant of the incomplete hyperbolic
structures on a 1–cusped finite volume hyperbolic 3–manifold
,
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
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
. Conjecturally
this new invariant is intimately related to the boundary of the hyperbolic Dehn surgery
space of .
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Keywords
hyperbolic cone-manifolds, character variety, ortholengths
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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
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