A new invariant on hyperbolic Dehn surgery space

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 2000

Primary: 57M50

Secondary: 57M27

References

Forward citations

Publication

Received: 24 October 2001

Revised: 24 May 2002

Accepted: 6 June 2002

Published: 22 June 2002

Authors

James G Dowty
Department of Mathematics University of Melbourne Parkville, 3052 Australia

Volume 2, issue 1 (2002)

James G Dowty