Geodesic currents of coarse negative curvature

TL;DR

This paper studies geodesic currents in coarse negative curvature spaces, proving density of strongly hyperbolic currents, contrasting with non-dense non-positively curved currents, and explores their geometric and spectral properties.

Contribution

It introduces the concept of strong hyperbolicity for geodesic currents, proves their density, and constructs new examples of strongly hyperbolic metrics not arising from CAT(0) spaces.

Findings

Strong hyperbolic currents are dense in the space of geodesic currents.

Currents from non-positively curved metrics are not dense.

Constructs infinitely many non-CAT(0) strongly hyperbolic metrics with distinct length spectra.

Abstract

Strong hyperbolicity is a coarse notion of negative curvature, stronger than Gromov hyperbolicity, that includes all CAT(-k) metrics for k positive and allows the use of dynamical techniques available in negative curvature, such as thermodynamical formalism. We prove that the subset of geodesic currents whose dual pseudometric is strongly hyperbolic is dense in the space of geodesic currents. The proof combines an elementary finite-cover argument with a characterization of strong hyperbolicity in terms of boundary data for pseudometrics dual to geodesic currents. In contrast, we show that currents arising from non-positively curved metrics on the surface are not dense. As a consequence, we construct infinitely many pairwise non-roughly-isometric invariant strongly hyperbolic geodesic metrics on the universal cover of the surface which are not CAT(0). Finally, we establish correlation counting results for the associated length spectra.