About ten years ago, Besson-Courtois-Gallot proved that a negatively curved rank one locally symmetric metric on a compact manifold uniquely minimizes volume growth entropy among all other metrics with the same volume. As usual, one would like to prove the same result in the finite volume case. Strong partial results were obtained by BolandConnell-Souto and the general case was proved recently. I’ll explain the background for this problem, and some of the ideas which go into the proof.