Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of L 1
[Noyaux CND presque invariants et actions propres uniformément Lipschitz sur des sous-espaces de L 1 ]
Vergara, Ignacio1
1 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central 9170020 (Chile)
Annales de l'Institut Fourier, Online first, 23 p.

We define the notion of almost invariant conditionally negative definite kernel and use it to give a characterisation of groups admitting a proper uniformly Lipschitz affine action on a subspace of an L 1 space. We show that this condition is satisfied by groups acting properly on products of quasi-trees, weakly amenable groups with Cowling–Haagerup constant 1, and a-TTT-menable groups.

Nous définissons la notion de noyau conditionnellement défini négatif presque invariant et nous l’utilisons pour donner une caractérisation des groupes admettant une action affine propre uniformément Lipschitz sur un sous-espace d’un espace L 1 . Nous montrons que cette condition est satisfaite par les groupes agissant proprement sur des produits de quasi-arbres, par les groupes faiblement moyennables avec constante de Cowling–Haagerup égale à 1 et par les groupes a-TTT-menables.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3731
Classification : 22D55, 43A35, 20F67, 46E30
Keywords: uniformly Lipschitz affine actions, subspaces of $L^1$, conditionally negative definite kernels, products of quasi-trees, weakly amenable groups, a-TTT-menable groups
Mots-clés : actions affines uniformément Lipschitz, sous-espaces de $L^1$, noyaux conditionnellement définis négatifs, produits de quasi-arbres, groupes faiblement moyennables, groupes a-TTT-menables

Vergara, Ignacio 1

1 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central 9170020 (Chile) 
@unpublished{AIF_0__0_0_A19_0,
     author = {Vergara, Ignacio},
     title = {Almost invariant {CND} kernels and proper uniformly {Lipschitz} actions on subspaces of $L^1$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3731},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Vergara, Ignacio
TI  - Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$
JO  - Annales de l'Institut Fourier
PY  - 2025
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3731
LA  - en
ID  - AIF_0__0_0_A19_0
ER  - 
%0 Unpublished Work
%A Vergara, Ignacio
%T Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$
%J Annales de l'Institut Fourier
%D 2025
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3731
%G en
%F AIF_0__0_0_A19_0
Vergara, Ignacio. Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$. Annales de l'Institut Fourier, Online first, 23 p.

[1] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008 | DOI | MR | Zbl

[2] Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji Proper actions on finite products of quasi-trees, Ann. Henri Lebesgue, Volume 4 (2021), pp. 685-709 | DOI | Numdam | MR | Zbl

[3] Bożejko, Marek; Picardello, Massimo A. Weakly amenable groups and amalgamated products, Proc. Am. Math. Soc., Volume 117 (1993) no. 4, pp. 1039-1046 | DOI | MR | Zbl

[4] Bretagnolle, Jean; Dacunha-Castelle, Didier; Krivine, Jean-Louis Lois stables et espaces L p , Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B, Volume 2 (1966), pp. 231-259 | Numdam | MR | Zbl

[5] Brown, Nathanial P.; Ozawa, Narutaka C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, 2008 | DOI | MR | Zbl

[6] Chatterji, Indira; Dahmani, François; Haettel, Thomas; Lécureux, Jean Tangent bundles of hyperbolic spaces and proper affine actions on L p spaces (2019) | arXiv | Zbl

[7] Chatterji, Indira; Druţu, Cornelia; Haglund, Frédéric Kazhdan and Haagerup properties from the median viewpoint, Adv. Math., Volume 225 (2010) no. 2, pp. 882-921 | DOI | MR | Zbl

[8] Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain Groups with the Haagerup property. Gromov’s a-T-menability, Progress in Mathematics, 197, Birkhäuser, 2001 | DOI | MR | Zbl

[9] de Cornulier, Yves; Stalder, Yves; Valette, Alain Proper actions of wreath products and generalizations, Trans. Am. Math. Soc., Volume 364 (2012) no. 6, pp. 3159-3184 | DOI | MR | Zbl

[10] de Cornulier, Yves; Tessera, Romain; Valette, Alain Isometric group actions on Hilbert spaces: growth of cocycles, Geom. Funct. Anal., Volume 17 (2007) no. 3, pp. 770-792 | DOI | MR | Zbl

[11] Druţu, Cornelia; Mackay, John M. Actions of acylindrically hyperbolic groups on ℓ 1 (2023) | arXiv | Zbl

[12] Gal, Światosław R.; Januszkiewicz, Tadeusz New a-T-menable HNN-extensions, J. Lie Theory, Volume 13 (2003) no. 2, pp. 383-385 | MR | Zbl

[13] Ghys, Étienne; de la Harpe, Pierre Espaces métriques hyperboliques, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) (Progress in Mathematics), Volume 83, Birkhäuser, 1990, pp. 27-45 | DOI | MR

[14] Gromov, Mikhail Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | MR | Zbl

[15] Guentner, Erik; Higson, Nigel Weak amenability of CAT (0)-cubical groups, Geom. Dedicata, Volume 148 (2010), pp. 137-156 | DOI | MR | Zbl

[16] Haagerup, Uffe An example of a nonnuclear C * -algebra, which has the metric approximation property, Invent. Math., Volume 50 (1978/79) no. 3, pp. 279-293 | DOI | MR | Zbl

[17] Han, Suzhen; Nguyen, Hoang Thanh; Yang, Wenyuan Property ( QT ) for 3-manifold groups, Algebr. Geom. Topol., Volume 25 (2025) no. 1, pp. 107-159 | DOI | MR | Zbl

[18] Julg, Pierre; Valette, Alain K-theoretic amenability for SL 2 (ℚ p ), and the action on the associated tree, J. Funct. Anal., Volume 58 (1984) no. 2, pp. 194-215 | DOI | MR | Zbl

[19] Kerr, Alice Tree approximation in quasi-trees, Groups Geom. Dyn., Volume 17 (2023) no. 4, pp. 1193-1233 | DOI | MR | Zbl

[20] Knudby, Søren Semigroups of Herz-Schur multipliers, J. Funct. Anal., Volume 266 (2014) no. 3, pp. 1565-1610 | DOI | MR | Zbl

[21] Kotowski, Marcin; Kotowski, Michał Random groups and property (T): Żuk’s theorem revisited, J. Lond. Math. Soc. (2), Volume 88 (2013) no. 2, pp. 396-416 | DOI | MR | Zbl

[22] de Laat, Tim; de la Salle, Mikael Actions of higher rank groups on uniformly convex Banach spaces (2023) | arXiv | Zbl

[23] Manning, Jason Fox Quasi-actions on trees and property (QFA), J. Lond. Math. Soc. (2), Volume 73 (2006) no. 1, pp. 84-108 | DOI | MR | Zbl

[24] Mizuta, Naokazu A Bożejko–Picardello type inequality for finite-dimensional CAT (0) cube complexes, J. Funct. Anal., Volume 254 (2008) no. 3, pp. 760-772 | DOI | MR | Zbl

[25] Nguyen, Hoang Thanh; Yang, Wenyuan Croke–Kleiner admissible groups: property (QT) and quasiconvexity, Mich. Math. J., Volume 73 (2023) no. 5, pp. 971-1019 | DOI | MR | Zbl

[26] Niblo, Graham; Reeves, Lawrence Groups acting on CAT (0) cube complexes, Geom. Topol., Volume 1 (1997), pp. 1-7 | DOI | MR | Zbl

[27] Nishikawa, Shintaro Sp(n,1) admits a proper 1-cocycle for a uniformly bounded representation (2022) | arXiv

[28] Nowak, Piotr W. Group actions on Banach spaces and a geometric characterization of a-T-menability, Topology Appl., Volume 153 (2006) no. 18, pp. 3409-3412 | DOI | MR | Zbl

[29] Nowak, Piotr W. Group actions on Banach spaces, Handbook of group actions. Vol. II (Advanced Lectures in Mathematics), Volume 32, International Press, 2015, pp. 121-149 | MR

[30] Ozawa, Narutaka Quasi-homomorphism rigidity with non-commutative targets, J. Reine Angew. Math., Volume 655 (2011), pp. 89-104 | DOI | MR | Zbl

[31] Ozawa, Narutaka Examples of groups which are not weakly amenable, Kyoto J. Math., Volume 52 (2012) no. 2, pp. 333-344 | DOI | MR | Zbl

[32] Ozawa, Narutaka; Popa, Sorin On a class of II 1 factors with at most one Cartan subalgebra, Ann. Math. (2), Volume 172 (2010) no. 1, pp. 713-749 | DOI | MR | Zbl

[33] de la Salle, Mikael Towards strong Banach property (T) for SL (3,ℝ), Isr. J. Math., Volume 211 (2016) no. 1, pp. 105-145 | DOI | MR | Zbl

[34] Schoenberg, Isaac J. Metric spaces and completely monotone functions, Ann. Math. (2), Volume 39 (1938) no. 4, pp. 811-841 | DOI | MR | Zbl

[35] Schoenberg, Isaac J. Metric spaces and positive definite functions, Trans. Am. Math. Soc., Volume 44 (1938) no. 3, pp. 522-536 | DOI | MR | Zbl

[36] Vergara, Ignacio Hyperbolicity and uniformly Lipschitz affine actions on subspaces of L 1 , Bull. Lond. Math. Soc., Volume 55 (2023) no. 5, pp. 2446-2455 | DOI | MR | Zbl

[37] Vergara, Ignacio Property (T) for uniformly bounded representations and weak*-continuity of invariant means (2023) | arXiv

[38] Watatani, Yasuo Property T of Kazhdan implies property FA of Serre, Math. Jap., Volume 27 (1982) no. 1, pp. 97-103 | MR | Zbl

[39] Willett, Rufus Some notes on property A, Limits of graphs in group theory and computer science (Fundamental Sciences. Mathematics), EPFL Press, 2009, pp. 191-281 | MR | Zbl

[40] Żuk, Andrzej Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal., Volume 13 (2003) no. 3, pp. 643-670 | DOI | MR | Zbl

Cité par Sources :