We construct Patterson–Sullivan measure and a natural metric on the unit space of a hyperbolic groupoid. In particular, this gives a new approach to defining SRB measures on Smale spaces and Anosov flows using Gromov hyperbolic graphs. Other results include a duality theory for hyperbolic groupoids graded by Hölder continuous Busemann cocycles, and a characterization of visual metrics.
Cite this article
Volodymyr V. Nekrashevych, Hyperbolic groupoids: metric and measure. Groups Geom. Dyn. 8 (2014), no. 3, pp. 883–932DOI 10.4171/GGD/252