Periodic Reporting for period 4 - INDEX (Rigidity of groups and higher index theory)
Okres sprawozdawczy: 2021-02-01 do 2022-07-31
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).
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.
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.
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.