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Dynamics, Geometry and Groups Seminar

Speaker: Abdul Zalloum (¾ÅÐãÖ±²¥)

Title: Constructing Isometric Actions on Metric Spaces from Walls

Abstract:
A guiding principle in geometric group theory is that one can learn about groups via their isometric actions on metric spaces. This approach is exceptionally successful when the underlying metric spaces satisfy certain non-positive curvature conditions. The following theorem is a simple example of the above phenomenon "a group is free if and only if it admits a free action on some tree". In this theorem, the geometric property of the tree having no loops, implied the algebraic property of the group being free and vice versa; this is a theme in geometric group theory. The goal of the talk is to discuss a procedure that takes as an input an arbitrary set S, a collection W of bi-partitions on S (thought of as "walls") and, depending on the combinatorics of W, it produces a range of metric spaces of non-positive curvature. The procedure is canonical in the sense that if G is any group acting on the set S preserving the collection of walls W, then it will act by isometries on the resulting metric spaces. I will also discuss some applications. This work is joint with Petyt.