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

Abstract

Let χ be the space of type-preserving SL(2, ℂ) characters of the punctured torus T. The Bowditch space χBQ is the largest open subset of χ 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] ∈ int(χ \ χBQ) if there exists an essential simple closed curve X on T such that |tr ρ(X)| < 0.5.