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Rigidity of groups and higher index theory

Periodic Reporting for period 3 - INDEX (Rigidity of groups and higher index theory)

Reporting period: 2019-08-01 to 2021-01-31

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).
The PI together with the team assembled for the project, consisting of three postdoctoral researchers and 1 PhD student, have obtained results on the topics outlined in the proposal. These results are documented in 9 mathematical articles, out of which 3 are already published in peer-reviewed international journal 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 two international workshops at the host institution
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 has been a long-standing open question. Our methods make use of convex optimization through a recently invented method of establishing property (T).
Our main progress beyond the state of the art 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. This identifies these groups as a potential counterexample to the Baum-Connes conjecture or some of its versions. Until the end of the project we expect to be able to provide new counterexamples to some versions of the Baum-Connes conjecture by constructing new ghost projections in crossed product algebras. We expect that property (T) and its stronger versions will play a fundamental role in this investigation.