  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 $|{\rm tr} \rho(X)|<0.5$.
