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 ]
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 p1 tels que chaque action par isométries affines de G sur 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 1p c , ou [1,p c ], [1,p c ]{2} pour un certain 1p 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 p is defined to be the set of p1 such that every action by affine isometries of G on 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 1p c , or [1,p c ], [1,p c ]{2} for some 1p 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 : https://doi.org/10.5802/aif.3348
Classification : 22D10,  22D12,  20CXX
Mots clés : Groupes avec la propriété (T), représentations orthogonales sur les espaces L p
@article{AIF_2021__71_1_1_0,
     author = {Lavy, Omer and Olivier, Baptiste},
     title = {Fixed-point spectrum for group actions by affine isometries on $L_{p}$-spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1--26},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {1},
     year = {2021},
     doi = {10.5802/aif.3348},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3348/}
}
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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