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Groups, Dynamics, and Approximation

Periodic Reporting for period 4 - GrDyAp (Groups, Dynamics, and Approximation)

Reporting period: 2021-04-01 to 2022-03-31

This ERC Consolidator Grant studied question of basic importance to the development of mathematics. We were trying to understand fundamental problems in the theory of groups, their dynamics and the corresponding problems of approximation by finite structures. The concept of group in the sense of mathematics captures the essence of symmetry which is fundamental to our understanding of nature, including physics, chemistry, biology. The mathematical side is group theory, the theoretical understanding of symmetries in general. It is especially intruiging to understand infinite symmetry groups.

The importance to society lies in the further development of the mathematical foundations of the sciences in general. The usefulness of mathematics in order to describe nature has been proves beyond any doubt over the last centuries. The development of the foundations of mathematics must therefore be a key interest for a society that values scientific progress.

The objectives were to clarify various approaches to infinite symmetry groups. Is very group sofic, i.e. does it admit a sofic approximation in the sense of Gromov and Weiss? This is still one of the most tantalyzing open problems in Geometric Group Theory, while we could make major contribution in understanding problems closely related to it. The ERC Consolidator Grant has extended the understanding of approximation problems, answered many questions related to the problem and brought forward new questions and problems that have inspired young researchers throughout the world.
Within the ERC Consolidator grant, we worked on numerous particular projects that all fit with the scheme of understanding infinite symmetry groups by some sort of approximation or representations as the group governing the dynamics of some particular action on a more concrete geometric object. This includes

- the work with Anton Claußniter on p-adic operator algebras, were we developed some analogues to the classical Hilbert space theory
- the work with Martin Schneider on amenability of large topological groups, were finitary approximation arises from the existence of Folner sets
- isometrisability and unitarisablity of discrete group (joint work with Maria Gerasimova, Nicolas Monod (Lausanne) and Dominik Gruber (Zürich))
- the non-existence of sofic approximation of certain actions, joint work with Gabor Kun (Budapest)
- joint work with Martin Nitsche on solvability of group equations for approximable groups
- the work with Henry Bradford (Göttingen) on laws for finite groups
- the work with Jakob Schneider on approximation properties with respect to all finite groups and also on properties of word maps on finite simple groups
- the work with Marcus de Chiffre, Oren Becker (Jerusalem), Alex Lubotzky (Jerusalem), Lev Glebsky (Mexico) and Narutaka Ozawa (Kyoto) on stability of amenable groups and the relationship between stability and higher Kazhdan properties
work with Jakob Schneider on word maps with constants in symmetric groups
- work with Bodirsky and Kummer on completely positive maps on rings of Hahn series over abelian valuation groups

These are just examples of questions that we tried to answer and in fact answered during the last 5 years.

Overall, I consider the project a success. We extended the knowledge in various areas of pure mathematics well beyond the state-of-the-art and many interesting results have come to light. I enjoyed working with excellent young mathematicians and post-docs and Dresden is now an established center for group theory and dynamics.
We made substantial progress during the duration of the ERC Consolidator Grant. Numerous contributions can be listed. It is in the nature of mathematical research, that it is hard to plan and hard to predict, on the other side it is a creative process that brings up surprising and deep connections that nobody envisioned when planning the project. In particular, I want to mention the connection between asymptotic properties of Cheeger constants and the Dixmier’s unitarisability problem (as explained in the paper with Gerasimova and Monod) or the surprising connections between Lie theory and properties of finite simple groups arising in the answer of a question Pillay (as explained in the work with Nikolov and J. Schneider). The work with M. Schneider has explored a new understanding of amenability for topological groups and highlighted connections to random walks on discrete groups through a detailed study of the Poisson boundary.
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