On purely loxodromic actions
Abstract
We construct an example of an isometric action of F(a,b) on a δ-hyperbolic graph Y, such that this action is acylindrical, purely loxodromic, has asymptotic translation lengths of nontrivial elements of F(a,b) separated away from 0, has quasiconvex orbits in Y, but such that the orbit map F(a,b) Y is not a quasi-isometric embedding.
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