Geometric Group Therapy Seminar
Speaker: Anders Karlsson (Université de Genève; Uppsala University)
Title: A CAT(0) alternative for amenable groups and a Kazhdan-type rigidity principle
Abstract: In a joint work with H. Izeki, R. Ji and Y. Wu, we establish an alternative for finitely generated amenable groups acting on CAT(0) spaces: either every action on a geodesically complete space of bounded geometry has a global fixed point, or the group admits a fixed-point-free action on a Euclidean space.
One ingredient of the proof is a Kazhdan-type rigidity principle, showing that for groups with the Euclidean fixed point property, the existence of almost fixed points forces an actual fixed point. The proof involves random walks.
As a consequence, finitely generated amenable torsion groups cannot act without a fixed point on geodesically complete CAT(0) spaces of bounded geometry; in visibility spaces, this conclusion holds without assuming amenability.