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INDEX SIGNED

Rigidity of groups and higher index theory

Total Cost €

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EC-Contrib. €

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Partnership

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Project "INDEX" data sheet

The following table provides information about the project.

Coordinator
INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK 

Organization address
address: UL. SNIADECKICH 8
city: WARSZAWA
postcode: 00 956
website: http://www.impan.gov.pl

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Poland [PL]
 Total cost 880˙625 €
 EC max contribution 880˙625 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2015-STG
 Funding Scheme ERC-STG
 Starting year 2016
 Duration (year-month-day) from 2016-08-01   to  2021-07-31

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK PL (WARSZAWA) coordinator 880˙625.00

Map

 Project objective

The Atiyah-Singer index theorem was one of the most spectacular achievements of mathematics in the XXth century, connecting the analytic and topological properties of manifolds. The Baum-Connes conjecture is a hugely successful approach to generalizing the index theorem to a much broader setting. It has remarkable applications in topology and analysis. For instance, it implies the Novikov conjecture on the homotopy invariance of higher signatures of a closed manifold and the Kaplansky-Kadison conjecture on the existence of non-trivial idempotents in the reduced group C*-algebra of a torsion-free group. At present, the Baum-Connes conjecture is known to hold for a large class of groups, including groups admitting metrically proper isometric actions on Hilbert spaces and Gromov hyperbolic groups.

The Baum-Connes conjecture with certain coefficients is known to fail for a class of groups, whose Cayley graphs contain coarsely embedded expander graphs. Nevertheless, the conjecture in full generality remains open and there is a growing need for new examples of groups and group actions, that would be counterexamples to the Baum-Connes conjecture. The main objective of this project is to exhibit such examples.

Our approach relies on strengthening Kazhdan’s property (T), a prominent cohomological rigidity property, from its original setting of Hilbert spaces to much larger classes of Banach spaces. Such properties are an emerging direction in the study of cohomological rigidity and are not yet well-understood. They lie at the intersection of geometric group theory, non-commutative geometry and index theory. In their study we will implement novel approaches, combining geometric and analytic techniques with variety of new cohomological constructions.

 Publications

year authors and title journal last update
List of publications.
2019 Yeong Chyuan Chung, Kang Li
Structure and K-theory of â„“p uniform Roe algebras
published pages: , ISSN: , DOI:
arxiv preprint 2019-10-03
2019 Kang Li, Piotr Nowak, Ján Špakula, Jiawen Zhang
Quasi-local Algebras and Asymptotic Expanders
published pages: , ISSN: , DOI:
arXiv preprint 2019-10-03
2019 Bruno de Mendonça Braga, Yeong Chyuan Chung, Kang Li
Coarse Baum-Connes conjecture and rigidity for Roe algebras
published pages: , ISSN: , DOI:
arXiv preprint 2019-10-03
2017 Marek Kaluba, Piotr W. Nowak, Narutaka Ozawa
Aut(F_5) has property (T)
published pages: , ISSN: , DOI:
arxiv preprint 2019-07-08
2018 Yeong Chyuan Chung, Kang Li
Rigidity of â„“p Roe-type algebras
published pages: 1056-1070, ISSN: 0024-6093, DOI: 10.1112/blms.12201
Bulletin of the London Mathematical Society 50/6 2019-07-08
2018 Marek Kaluba, Dawid Kielak, Piotr W. Nowak
On property (T) for Aut(F_n) and SL_n(Z)
published pages: , ISSN: , DOI:
arxiv preprint 2019-04-18
2018 Yeong Chyuan Chung
Dynamical Complexity and K-Theory of Lp Operator Crossed Products
published pages: , ISSN: , DOI:
arxiv preprint 2019-04-18
2018 Yongle Jiang, Piotr W. Nowak
Singular subgroups in $widetilde{A}_2$-groups and their von Neumann algebras
published pages: , ISSN: , DOI:
arxiv preprint 2019-04-18
2018 Yeong Chyuan Chung, Piotr W. Nowak
Expanders are counterexamples to the coarse $ell_p$-Baum-Connes conjecture
published pages: , ISSN: , DOI:
arxiv preprint 2019-04-18
2018 Li, Kang; Wang, Zhijie; Zhang, Jiawen
A quasi-local characterisation of $L^p$-Roe algebras
published pages: , ISSN: , DOI:
arxiv preprint 1 2019-04-18

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