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

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)|