No. 677120 INDEX

Rigidity of groups and higher index theory

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.

Aut (F_5) has property (T)

Author(s): Marek Kaluba, Piotr W. Nowak, Narutaka Ozawa

Published in: Mathematische Annalen, 2019, ISSN 0025-5831

Publisher: Springer Verlag

DOI: 10.1007/s00208-019-01874-9

On property (T) for Aut(F_n) and SL_n(Z)

Author(s): Marek Kaluba, Dawid Kielak, Piotr W. Nowak

Published in: Annals of Mathematics, Volume 193 (2021), Issue 2, 2018, Page(s) 539-562, ISSN 0003-486X

Publisher: Princeton University Press

DOI: 10.4007/annals.2021.193.2.3

Asymptotic expansion in measure and strong ergodicity

Author(s): Kang Li; Federico Vigolo; Jiawen Zhang

Published in: Journal of Topology and Analysis, 14, 2021, ISSN 1793-5253

Publisher: World Scientific Publishing Co. Pte Ltd

DOI: 10.1142/s1793525321500278

An algebraic characterization of ample type I groupoids

Author(s): Favre, Gabriel; Raum, Sven

Published in: Semigroup Forum, 104 (2022), no. 1, 2022, Page(s) 58–71, ISSN 0037-1912

Publisher: Springer Verlag

DOI: 10.1007/s00233-021-10241-7

The Haagerup property for twisted groupoid dynamical systems

Author(s): Bartosz K. Kwaśniewski; Kang Li; Adam Skalski

Published in: Journal of Functional Analysis, Volume 283, Issue 1, 1 July 2022, 109484, 2022, ISSN 0022-1236

Publisher: Academic Press

DOI: 10.1016/j.jfa.2022.109484

Measured expanders

Author(s): Kang Li; Ján Špakula; Jiawen Zhang

Published in: Journal of Analysis and Topology, 31, 2022, ISSN 1793-5253

Publisher: World Scientific Publishing Co. Pte Ltd

DOI: 10.48550/arxiv.2104.06052

Group algebra criteria for vanishing of cohomology

Author(s): Uri Bader, Piotr W. Nowak

Published in: Journal of Functional Analysis, 279/11, 2020, Page(s) 108730, ISSN 0022-1236

Publisher: Academic Press

DOI: 10.1016/j.jfa.2020.108730

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Authors: Sven Raum; Adam Skalski

Published in: Advances in Mathematics, Volume 407, 8 October 2022, 108559, 2022, ISSN 0001-8708

Publisher: Academic Press

DOI: 10.48550/arxiv.2103.02962

Non-orderability of random triangular groups by using random 3CNF formulas

Author(s): Damian Orlef

Published in: Bulletin of the London Mathematical Society, Vol. 53, issue 5, 2021, ISSN 0024-6093

Publisher: Oxford University Press

DOI: 10.48550/arxiv.2102.01601

Coarse Baum-Connes conjecture and rigidity for Roe algebras

Author(s): Bruno M. Braga, Yeong Chyuan Chung, Kang Li

Published in: Journal of Functional Analysis, 279/9, 2020, Page(s) 108728, ISSN 0022-1236

Publisher: Academic Press

DOI: 10.1016/j.jfa.2020.108728

Dynamic asymptotic dimension for actions of virtually cyclic groups

Author(s): Massoud Amini; Kang Li; Damian Sawicki; Ali Shakibazadeh

Published in: Proceedings of the Edinburgh Mathematical Society, Volume 64 , Issue 2 , May 2021, 2021, Page(s) 364 - 372, ISSN 0013-0915

Publisher: Cambridge University Press

DOI: 10.1017/s0013091521000183

The type semigroup, comparison and almost finiteness for ample groupoids

Author(s): Pere Ara; Christian Bönicke; Joan Bosa; Kang Li

Published in: Ergodic Theory and Dynamical Systems, 1, 2021, ISSN 0143-3857

Publisher: Cambridge University Press

DOI: 10.48550/arxiv.2001.00376

On complements of Kazhdan projections in semisimple groups

Author(s): Piotr W. Nowak, Eric Reckwerdt

Published in: Studia Mathematica, 246/1, 2019, Page(s) 63-69, ISSN 0039-3223

Publisher: Polska Akademia Nauk

DOI: 10.4064/sm171126-18-7

A quasi-local characterisation of $L^p$-Roe algebras

Author(s): Li, Kang; Wang, Zhijie; Zhang, Jiawen

Published in: Journal of Mathematical Analysis and Applications, Volume 474, Issue 2, 15 June 2019, 2019, Page(s) 1213-1237, ISSN 0022-247X

Publisher: Academic Press

DOI: 10.1016/j.jmaa.2019.02.013

Rigidity of p Roe-type algebras

Author(s): Yeong Chyuan Chung, Kang Li

Published in: Bulletin of the London Mathematical Society, 50/6, 2018, Page(s) 1056-1070, ISSN 0024-6093

Publisher: Oxford University Press

DOI: 10.1112/blms.12201

Dynamical Complexity and K-Theory of Lp Operator Crossed Products

Author(s): Yeong Chyuan Chung

Published in: Journal of Topology and Analysis, 13 (2021), no. 3, 2021, Page(s) 809–841, ISSN 1793-5253

Publisher: World Scientific Publishing Co. Pte Ltd

DOI: 10.1142/s1793525320500314

Structure and K-theory of p uniform Roe algebras

Author(s): Yeong Chyuan Chung, Kang Li

Published in: J. Noncommut. Geom., 15 (2021), no. 2, 2021, Page(s) 581–614, ISSN 1661-6952

Publisher: European Mathematical Society Publishing House

DOI: 10.4171/jncg/405

Quasi-local Algebras and Asymptotic Expanders

Author(s): Kang Li, Piotr W. Nowak, Ján Špakula, Jiawen Zhang

Published in: Groups Geom. Dyn., 15 (2021), no. 2, 2021, Page(s) 655–682, ISSN 1661-7207

Publisher: European Mathematical Society Publishing House

DOI: 10.4171/ggd/610

Certifying numerical estimates of spectral gaps

Author(s): Marek Kaluba, Piotr W. Nowak

Published in: Groups Complexity Cryptology, 0/0, 2018, ISSN 1867-1144

Publisher: Walter de Gruyter GmbH & Co. KG

DOI: 10.1515/gcc-2018-0004

On the structure of asymptotic expanders

Author(s): Ana Khukhro; Kang Li; Federico Vigolo; Jiawen Zhang

Published in: Advances in Mathematics, 6, 2021, ISSN 0001-8708

Publisher: Academic Press

DOI: 10.48550/arxiv.1910.13320

Strict comparison for C*-algebras arising from almost finite groupoids

Author(s): Pere Ara; Christian Bönicke; Joan Bosa; Kang Li

Published in: Banach Journal of Mathematical Analysis, volume 14, pages1692–1710, 2020, ISSN 1735-8787

Publisher: Banach Mathematical Research Group

DOI: 10.48550/arxiv.2002.12221

Amenability, proximality and higher order syndeticity

Author(s): Kennedy, Matthew; Raum, Sven; Salomon, Guy

Published in: Forum Math. Sigma, 10 (2022), 2022, Page(s) Paper No. e22, 28 pp, ISSN 2050-5094

Publisher: Cambridge Univ. Press

DOI: 10.1017/fms.2022.11

S^1-bounded Fourier multipliers on H^1() and functional calculus for semigroups

Author(s): Loris Arnold, Christian Le Merdy, Safoura Zadeh

Published in: preprint, 2022

Publisher: arxiv

DOI: 10.48550/arxiv.2203.16829

BEHAVIOR OF KREISS BOUNDED C0-SEMIGROUPS ON A HILBERT SPACE

Author(s): Loris Arnold

Published in: preprint, 2022

Publisher: arxiv

DOI: 10.48550/arxiv.2206.10426

The ideal intersection property for essential groupoid C*-algebras

Author(s): Kennedy, Matthew; Kim, Se-Jin; Li, Xin; Raum, Sven; Ursu, Dan

Published in: preprint, submitted, 2021

Publisher: arxiv

DOI: 10.48550/arxiv.2107.03980

Measured asymptotic expanders and rigidity for Roe algebras

Author(s): Kang Li, Ján Špakula, Jiawen Zhang

Published in: preprint, submitted, 2022

Publisher: arxiv

DOI: 10.48550/arxiv.2010.10749

Spectral gap for the cohomological Laplacian of SL_3(Z)

Author(s): Marek Kaluba, Piotr Mizerka, Piotr W. Nowak

Published in: preprint, 2022

Publisher: arxiv

DOI: 10.48550/arxiv.2207.02783

Higher Kazhdan projections, ℓ2-Betti numbers and Baum-Connes conjectures

Author(s): Kang Li, Piotr W. Nowak, Sanaz Pooya

Published in: arxiv preprint, 2019

Publisher: arxiv

Expanders are counterexamples to the coarse $\ell_p$-Baum-Connes conjecture

Author(s): Yeong Chyuan Chung, Piotr W. Nowak

Published in: arxiv preprint, 2018

Publisher: arxiv

Coarse Baum-Connes conjecture and rigidity for Roe algebras

Author(s): Bruno de Mendonça Braga, Yeong Chyuan Chung, Kang Li

Published in: arXiv preprint, 2019

Publisher: arxiv

From the coarse geometry of warped cones to the measured coupling of groups

Author(s): Das, Kajal

Published in: preprint, submitted, 2020

Publisher: arxiv

DOI: 10.48550/arxiv.2004.10000

A Markovian and Roe-algebraic approach to asymptotic expansion in measure

Author(s): Li, Kang; Vigolo, Federico; Zhang, Jiawen

Published in: preprint, submitted, 2020

Publisher: arxiv

DOI: 10.48550/arxiv.2008.12572