The SL(2,C) character variety of the one-holed torus

Abstract

In this note we announce several results concerning the SL(2,C) character variety $\mathcal{X}$ of the one-holed torus. We give a description of the largest open subset ${\mathcal{X}}_{BQ}$ of $\mathcal{X}$ on which the mapping class group $\Gamma$ acts properly discontinuously, in terms of two very simple conditions, and show that a series identity generalizing McShane's identity for the punctured torus holds for all characters in this subset. We also give variations of the McShane-Bowditch identities to characters fixed by an Anosov element of $\Gamma$ with applications to closed hyperbolic three manifolds. Finally we give a definition of end invariants for SL(2,C) characters and give a partial classification of the set of end invariants of a character in $\mathcal{X}$.

8 pages, announcement