no. 1
Fixed-point spectrum for group actions by affine isometries on L p -spaces
[Le spectre des points fixes pour les actions de groupes par isométries affines sur les espaces L p ]
Lavy, Omer1 ; Olivier, Baptiste2
1 Weizmann Institute of Science Department of Mathematics Herzl St 234, Rehovot (Israel)
2 Orange Labs 4 rue clos courtel 35510 Rennes (France)
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 1-26.

Le spectre des points fixes d’un groupe localement compact à base dénombrable G est défini comme l’ensemble des p≥1 tels que chaque action par isométries affines de G sur ℓ p admet un point fixe. Nous montrons que cet ensemble est soit vide, soit peut s’écrire sous une des formes suivantes : [1,p c [, [1,p c [∖{2} pour un certain 1≤p c ≤∞, ou [1,p c ], [1,p c ]∖{2} pour un certain 1≤p c <∞. Ce résultat est en lien étroit avec la conjecture de C. Drutu affirmant que le spectre des points fixes est un ensemble connexe pour les actions isométriques sur L p (0,1).

Plus généralement, nous étudions les propriétés topologiques du spectre des points fixes sur L p (X,μ) pour des espaces mesurés arbitraires (X,μ), et nous montrons des résultats partiels dans le sens de la conjecture pour les actions sur L p (0,1). En particulier, nous prouvons que le spectre associé aux actions dont la partie linéaire est π est soit vide, soit un intervalle de la forme [1,p c ] (p c ≥1) ou [1,∞[, dès que π est une représentation orthogonale associée à une action ergodique préservant la mesure sur un espace mesuré (X,μ) de mesure finie.

The fixed-point spectrum of a locally compact second countable group G on ℓ p is defined to be the set of p≥1 such that every action by affine isometries of G on ℓ p admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms: [1,p c [, [1,p c [∖{2} for some 1≤p c ≤∞, or [1,p c ], [1,p c ]∖{2} for some 1≤p c <∞. This result is closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions on L p (0,1).

More generally, we study the topological properties of the fixed-point spectrum on L p (X,μ) for general measure spaces (X,μ), and show partial results toward the conjecture for actions on L p (0,1). In particular, we prove that the spectrum associated with actions with linear part π is either empty, or an interval of the form [1,p c ] (p c ≥1) or [1,∞[, whenever π is an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space (X,μ).

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DOI : 10.5802/aif.3348
Classification : 22D10, 22D12, 20CXX
Keywords: Groups with property $(T)$, orthogonal representations on $L_p$-spaces
Mot clés : Groupes avec la propriété $(T)$, représentations orthogonales sur les espaces $L_p$
Lavy, Omer 1 ; Olivier, Baptiste 2

1 Weizmann Institute of Science Department of Mathematics Herzl St 234, Rehovot (Israel)
2 Orange Labs 4 rue clos courtel 35510 Rennes (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Lavy, Omer; Olivier, Baptiste. Fixed-point spectrum for group actions by affine isometries on $L_{p}$-spaces. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 1-26. doi : 10.5802/aif.3348. https://aif.centre-mersenne.org/articles/10.5802/aif.3348/

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