A new invariant on hyperbolic Dehn surgery space

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 $M$.