The complement of the Bowditch space in the SL(2,C) character variety

Abstract

Let $\mathcal{X}$ be the space of type-preserving $\SL(2,C)$ characters of the punctured torus $T$. The Bowditch space ${\mathcal{X}}_{BQ}$ is the largest open subset of $\mathcal{X}$ on which the mapping class group acts properly discontinuously, this is characterized by two simple conditions called the $BQ$-conditions. In this note, we show that $[\rho ]$ is in the interior of the complement of ${\mathcal{X}}_{BQ}$ if there exists an essential simple closed curve $X$ on $T$ such that $|\mathrm{tr}\rho (X)|<0.5$.

6 pages