Result: Strong Property (T), weak amenability and $\ell^p$-cohomology in $\tilde{A}_2$-buildings: Strong property (T), weak amenability and \(\ell^p\)-cohomology in \(\tilde{A}_2\)-buildings
Title:
Strong Property (T), weak amenability and $\ell^p$-cohomology in $\tilde{A}_2$-buildings: Strong property (T), weak amenability and \(\ell^p\)-cohomology in \(\tilde{A}_2\)-buildings
Authors:
Contributors:
de la Salle, Mikael, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse (EDPA), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Justus-Liebig-Universität Gießen = Justus Liebig University (JLU), ANR-19-CE40-0002,ANCG,Analyse non commutative sur les groupes et les groupes quantiques(2019), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), ANR-16-CE40-0022,AGIRA,Actions de Groupes, Isométries, Rigidité et Aléa(2016)
Source:
Annales Scientifiques de l'École Normale Supérieure.
Publication Status:
Preprint
Publisher Information:
Societe Mathematique de France, 2025.
Publication Year:
2025
Subject Terms:
weak amenability, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], \(\ell^{p}\)-cohomology, Group Theory (math.GR), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], undistorted lattice, 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Group actions on combinatorial structures, FOS: Mathematics, 0101 mathematics, Operator Algebras (math.OA), Kazhdan's property (T), the Haagerup property, and generalizations, strong property (T), [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], 20F65, 51E24, [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Mathematics - Operator Algebras, Embeddings of discrete metric spaces into Banach spaces, applications in topology and computer science, buildings, Buildings and the geometry of diagrams, [MATH.MATH-OA] Mathematics [math]/Operator Algebras [math.OA], Geometric group theory, Mathematics - Group Theory
Document Type:
Academic journal
Article
File Description:
application/xml
ISSN:
1873-2151
0012-9593
0012-9593
DOI:
10.24033/asens.2593
DOI:
10.48550/arxiv.2010.07043
Access URL:
Rights:
arXiv Non-Exclusive Distribution
Accession Number:
edsair.doi.dedup.....1e4d4f2af2c8af5973f88df5f623f621
Database:
OpenAIRE
Further Information
We prove that cocompact (and more generally: undistorted) lattices on $\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our methods also give two further results. First, we show that the first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for any finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in $[\frac 4 3,4]$ and the reduced $C^*$-algebra associated to an $\tilde{A}_2$-lattice do not have the operator space approximation property and, consequently, that the lattice is not weakly amenable.
v1: 68 pages, 6 figures; v2: 79 pages, many improvements in the presentation. To appear in Ann. Sci. \'Ecole Norm. Sup