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Geometry and Analysis of Group Rings

Final Report Summary - GEOMANGROUP (Geometry and Analysis of Group Rings)

The project „Geometry and Analysis of Group Rings“, funded as an ERC Starting Grant during the period from October 2011 until September 2016 was successful. Documented in over 50 publication in highly ranked journals, breakthrough achievements were obtained in the area of group theory, functional analysis, and the theory of dynamical systems. The ERC Starting Grant allowed Andreas Thom to develop his own research agenda and build up a successful research group including PhD students and postdoctoral researchers.
One of the main breakthroughs was a solution of a Conjecture of Deninger and a Conjecture of Lück about the entropy of algebraic actions of amenable groups and the vanishing of L2-torsion for amenable groups. These (at first sight) unrelated conjectures could be proved in joint work with Hanfeng Li from SUNY Buffalo using an astonishing combination of homological algebra and entropy theory. These results were published in the Journal of the AMS.
The understanding of the class of sofic groups and the classes of groups satisfying similar approximation properties could be deepened considerably. A relationship between the cost of a group and Dixmier’s Unitarisability Conjecture could be proved. In general, the project provided evidence that the multidisciplinary approach of Andreas Thom can yield substantial progress when applied to various longstanding problems.