Rigidity of groups and higher index theory

Project Information

INDEX

Grant agreement ID: 677120

DOI

10.3030/677120

Project closed

EC signature date 15 March 2016

Start date 1 August 2016

End date 31 July 2022

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 880 625,00

EU contribution

€ 880 625,00

Coordinated by

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

Periodic Reporting for period 4 - INDEX (Rigidity of groups and higher index theory)

Reporting period: 2021-02-01 to 2022-07-31

Summary of the context and overall objectives of the project

The general aim of the project is to find new examples of certain algebraic objects called groups that would have certain exotic properties. Namely, the groups cannot be realized as interesting groups of symmetries of the (possibly infinite-dimensional) Euclidean space. This property of groups is called property (T) and whenever it has several important applications in several fields of theoretical mathematics, including in graph theory, as well as computer science. The project is centered around finding groups with property (T) in order to identify counterexamples to a well-known problem in index theory, the Baum-Connes conjecture (and several of its versions).

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Throughout the project the team, consisting of the PI and 11 members have obtained results on the topics outlined in the proposal. These results are documented in over 30 mathematical articles, out of which already 25 are published or accepted in established, peer-reviewed international journals and 6 are submitted for publication.
Team members have travelled to conferences and seminars around the world to collaborate and to disseminate the results of the project; invited expert guest to visit the Host Institution, collaborate and present talks on their results; co-organized international workshops at the host institution; given several presentations on the results of the project in popular science media outlets.
The most significant results of the project to date is a series of results showing that Aut(F_n), the automorphism group of the free group on n generators, has Kazhdan's property (T) for n from 5 to infinity. This resolves been a long-standing open problem and has several applications, including some in theoretical computer science. Our methods make use of convex optimization through a recently invented method of establishing property (T) by solving equations in the group ring via semidefinite programming. Other results include new constructions of projections that give rise to interesting classes in K-theory. These classes are used to find counterexamples to certain versions of the Baum-Connes conjecture.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The main achievement beyond the state of the art of the project is a series of results showing that Aut(F_n), the automorphism group of the free group on n generators, has Kazhdan's property (T) for n from 5 to infinity. This result answers a long-standing open problem. These results identify these groups as a potential counterexample to the Baum-Connes conjecture or some of its versions. Another series of results is a construction of several new projections that give rise to K-theory classes that do not lie in the image of the Baum-Connes assembly map. These projections are related to the presence of spectral gaps and various forms of expansion. Such expansion properties are important from the point of view applications, such as in computer science.

Convergence of a semidefinite solver to a solution when proving property (T) for Aut(F_5)

Graphical depiction of a 4641x4641 matrix solving the equation proving property (T) for Aut(F_5)