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We introduce "co-geodesic currents," a family of G-invariant measures on hyperplanes at infinity for a Gromov hyperbolic group G, generalizing geodesic currents from surface groups. Co-geodesic currents naturally arise from geometric actions of G on CAT(0) cube complexes or certain actions on real trees. We establish a natural intersection pairing between co-geodesic and geodesic currents, extending Bonahon’s intersection number for surface groups. Moreover, we show that every non-trivial co-geodesic current defines a pseudo-metric space with a measured wall structure.
We deduce that the existence of certain non-trivial co-geodesic currents imply lack of property (T) or the Haagerup property for G.
This is joint work in progress with Eduardo Reyes.