Volume entropy of hyperbolic buildings.

Volume entropy of a Riemannian manifold is the exponential growth rate of the volumes of balls. Entropy rigidity for rank-1 Riemannian manifolds is known: a theorem of Besson-Courtois-Gallot says that the locally symmetric metrics attain minimal volume entropy among all Riemannian metrics. In this talk, we are interested in entropy for buildings, especially hyperbolic ones. We will give several characterizations of the volume entropy, analogous to the ones for trees, and show that Liouville measure is not entropy maximizing measure. As a result, we obtain a strict lower bound on volume entropy for certain hyperbolic buildings.