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 X of the one-holed torus. We give a description of the largest open subset XBQ of X on which the mapping class group Γ acts properly dis-continuously, 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 Γ with applications to closed hyperbolic three manifolds. Finally we give a definition of end invari-ants for SL(2,C) characters and give a partial classification of the set of end invariants of a character in X. 1