Mathematics – Geometric Topology
Scientific paper
2002-07-05
Algebr. Geom. Topol. 2 (2002) 465-497
Mathematics
Geometric Topology
Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-23.abs.html
Scientific paper
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.
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