We establish a correspondence between continuous reparameterizations of the Mineyev flow of a hyperbolic group and a natural space of coarse geometric hyperbolic metrics on the group.

This space includes many important Gromov hyperbolic metrics, such as metrics induced from co-bounded and proper actions on geodesic Gromov hyperbolic spaces or Green metrics from admissible random walks.

Our correspondence extends beyond the Hölder setting to arbitrary continuous reparameterizations. For instance, it produces continuous reparameterizations associated to natural geometric objects that do not admit Hölder reparameterizations, including cubulations, geodesic currents on surface groups, and word metrics. A key ingredient in our proof is the density of Green metrics in a moduli space of metrics equipped with the (extension of the) symmetrized Thurston metric.

This is joint work with Stephen Cantrell and Eduardo Reyes.