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Explore the third lecture in a series on median spaces and algebras, delivered by Elia Fioravanti as part of the Workshop on Cube Complexes and Combinatorial Geometry. Delve into the fascinating world of median spaces, which generalize L^1 spaces, CAT(0) cube complexes, and R-trees. Discover how these mathematical structures play crucial roles in geometric group theory, appearing in both finite and infinite-dimensional contexts. Examine the applications of finite-dimensional median spaces in studying quasi-flats, quasi-isometries, and automorphisms for groups with HHS structure and cocompactly cubulated groups. Investigate the use of infinite-dimensional median spaces and their dual concept, spaces with measured walls, in testing functional analytic properties like Kazhdan's property (T) and the Haagerup property. Gain insights into the connections between median spaces and median algebras, and understand their implications for cocompactly cubulated groups.
