Rigidity of groups and higher index theory

Projektinformationen

INDEX

ID Finanzhilfevereinbarung: 677120

DOI

10.3030/677120

Projekt abgeschlossen

EK-Unterschriftsdatum 15 März 2016

Startdatum 1 August 2016

Enddatum 31 Juli 2022

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 880 625,00

EU-Beitrag

€ 880 625,00

Koordiniert durch

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

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

Berichtszeitraum: 2021-02-01 bis 2022-07-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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).

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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)