Bowditch representations in Gromov-hyperbolic spaces

Bowditch, followed by Tan-Wong-Zhang, introduced in 1998 a class of representations of the once-punctured torus group into PSL(2,C). Using trace relations in PSL(2,C), they give a condition on a representation which ensures that the translation lengths of the images of simple closed curves grow linearly with respect to the word length. In this talk, I will explain how to define and study a generalization of these conditions in the context of Gromov-hyperbolic spaces, where no trace relations hold, and give several characterizations of this set. I will also explain how this work makes it possible to study the dynamics of the mapping class group on the space of Bowditch representations.