Strongly hyperbolic geodesic currents are dense in the space of geodesic currents, yielding infinitely many pairwise non-roughly-isometric strongly hyperbolic metrics on the universal cover that are not CAT(0).
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Geodesic currents of coarse negative curvature
Strongly hyperbolic geodesic currents are dense in the space of geodesic currents, yielding infinitely many pairwise non-roughly-isometric strongly hyperbolic metrics on the universal cover that are not CAT(0).