A Bishop-Gromov Type Inequality for some Metric Spaces

The work concerns a δ-hyperbolic metric space (X,d) (possibly with extra conditions) and the main assumption is that its entropy, denoted by H, is bounded above. Then, if a subgroup of its isometry group acts properly and co-compactly and if D denotes the diameter of the quotient, we will show a Bishop-Gromov type inequality on (X,d) only in terms of δ, H and D. It is a curvature-free inequality and we will explain how the bound on the entropy plays the role of a (weak version of a) lower bound on the Ricci curvature and how the δ-hyperbolicity relates to a bound on the negative part of the sectional curvature. Some consequences of this inequality are a finiteness theorem as well as a compactness result.

This is joint work with G. Courtois, S. Gallot and A. Sambusetti.