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Explore advanced geometric group theory through this mathematical seminar that examines the intricate relationships between asymptotic dimension, isoperimetric problems, and the Traveling Salesman Problem within group structures. Delve into the concept of Følner functions and their relationship to isodiametric functions, discovering when these functions are "lossless" and when they diverge in their asymptotic behavior. Investigate how the isodiametric function relates to control functions for asymptotic dimension and invariants connected to the Universal Traveling Salesman Problem across various group classes. Analyze specific examples including diagonal products of Brieussel–Zheng associated with expander graphs, wreath products, Hall's nilpotent-by-abelian groups, and Lampshuffler groups to understand when control functions maintain their lossless property despite non-lossless Følner functions. Examine the construction of amenable groups with non-lossless control functions and explore how isoperimetric problems influence fundamental algebraic properties of groups. Learn about Shalom's property HFD in amenable groups of finite Assouad–Nagata dimension and understand why such groups cannot be simple or purely torsion, contrasting with the simple groups of Burger and Mozes that are quasi-isometric to products of trees.
