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 $[ρ]$ is in the interior of the complement of ${\mathcal X}_{BQ}$ if there exists an essential simple closed curve $X$ on $T$ such that $|{\rm tr} ρ(X)|<0.5$.