#### Volume 2, issue 1 (2002)

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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
Primary: 57M50
Secondary: 57M27