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

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

2005-09-02

arxiv.org/abs/math/0509033v1

Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 103-110

Mathematics

Geometric Topology

8 pages, announcement

Scientific paper

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 ${\mathcal X}$.

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