These two talks will build on the ideas presented in Juan Souto's lectures on Mostow rigidity. Taking Mostow's theorem as a starting point, we will discuss some other notions of rigidity in the Riemannian setting, focusing primarily on the entropy-rigidity theorem of U. Hamenst\"adt: the metrics on a compact hyperbolic manifold with minimal volume entropy are precisely the hyperbolic metrics. From there, we will proceed toward metric geometry through the work of M. Bourdon and of M. Bonk and B. Kleiner, who adapted Hamenst\"adt's result to the metric setting. In particular, we will highlight the connection between these two settings; namely, group actions on $\textup{\text{CAT}}(-1)$ spaces. This will motivate, for the second talk, an excursion into coarse geometry, where one can formulate an analogous entropy-rigidity statement for group actions on Gromov hyperbolic spaces. We will then present the main steps in the proof of this statement.
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