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Groups, Dynamics, and Approximation

Periodic Reporting for period 3 - GrDyAp (Groups, Dynamics, and Approximation)

Reporting period: 2019-10-01 to 2021-03-31

This ERC Consolidator Grant studies question of basic importance to the development of mathematics. We are 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. A natural question arising in this situation is how infinite groups can be approximated in a suitable sense by finite groups. It is well-known that this is not possible in general and many leading mathematicians have suggested alternative and more flexible objects that could serve as finite templates that approximate the infinite symmetry group. The study of sofic approximations is exactly one possible promising approach.

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 are 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 one of the most tantalyzing open problems in Geometric Group Theory. It would be very interesting for applications to obtain a general positive answer, but also a negative answer would be useful, as it shows that limitations of this approach. We are currently trying to construct a non-sofic group. This is one of the main objectives for the remaining time of th ERC Consolidator Grant.
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 develop 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

These are just examples of questions that we tried to answer and in fact answered during the last 2,5 years.
It is hard to predict mathematical research, but our definite aim is to construct a non-sofic group and the work with Gabor Kun is a very good starting point for this. Moreover, I will try to deepen my understanding and the understanding of my group on Dixmier's unitarisability problem and make further progress on understanding metric ultraproducts of finite simple groups. Any progress in one of these fields would be groundbreaking and worth the effort.