Algebraic & Geometric Topology 2 (2002) 465–497

DOI: 10.2140/agt.2002.2.465

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

Primary: 57M50

Secondary: 57M27

References

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Publication

Received: 24 October 2001
Revised: 24 May 2002
Accepted: 6 June 2002
Published: 22 June 2002

Authors