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Explore the concept of combinatorial dimension and higher-rank hyperbolicity in this 57-minute lecture by Urs Lang. Delve into Dress's characterization of metric spaces with combinatorial dimension at most n using a 2(n+1)-point inequality. Examine a relaxed version of this inequality, termed (n,δ)-hyperbolicity, which generalizes Gromov's quadruple definition of δ-hyperbolicity. Learn about the properties of (n,δ)-hyperbolic spaces, including the slim (n+1)-simplex property. Discover connections to recent developments in geometric group theory, including applications to Helly groups and hierarchically hyperbolic groups. Based on joint work with Martina Jørgensen, this talk provides insights into the intersection of metric geometry and group theory.
