Rigidity of groups and higher index theory

Informacje na temat projektu

INDEX

Identyfikator umowy o grant: 677120

DOI

10.3030/677120

Projekt został zamknięty

Data podpisania przez KE 15 Marca 2016

Data rozpoczęcia 1 Sierpnia 2016

Data zakończenia 31 Lipca 2022

Finansowanie w ramach

• EXCELLENT SCIENCE - European Research Council (ERC)

Koszt całkowity

€ 880 625,00

Wkład UE

€ 880 625,00

Koordynowany przez

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

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

Okres sprawozdawczy: 2021-02-01 do 2022-07-31

Podsumowanie kontekstu i ogólnych celów projektu

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).

Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty

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.

Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu)

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)