The complement of the Bowditch space in the $\mathrm{SL}(2,\mathbb{C})$ character variety

Abstract

Let $\mathcal X$ be the space of type-preserving $\mathrm{SL}(2,\mathbb{C})$ characters of the punctured torus $T$. The Bowditch space $\mathcal{X}_{\mathrm{BQ}}$ 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 $[\rho] \in \operatorname{int}(\mathcal{X} \setminus \mathcal{X}_{\mathrm{BQ}})$ if there exists an essential simple closed curve $X$ on $T$ such that $|\tr\rho(X)|<0.5$.