For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.
