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Explore a mathematical lecture that delves into asymptotic rank and injective hulls, examining a coarse version of Dress' 2(n+1)-inequality for metric spaces of combinatorial dimension n or less. Learn about (n,δ)-hyperbolicity, which generalizes Gromov's δ-hyperbolicity definition when n=1, and discover how the ℓ∞ product of n δ-hyperbolic spaces exhibits (n,δ)-hyperbolicity. Understand the slim (n+1)-simplex property in (n,δ)-hyperbolic spaces and its parallel to quasi-geodesic triangles in Gromov hyperbolic spaces. Examine practical applications through examples demonstrating how Helly groups and hierarchically hyperbolic groups of asymptotic rank n act geometrically on (n,δ)-hyperbolic spaces, based on joint research with Urs Lang.
