The fixed-point spectrum of a locally compact second countable group on is defined to be the set of such that every action by affine isometries of on admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms: , for some , or , for some . This result is closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions on .
More generally, we study the topological properties of the fixed-point spectrum on for general measure spaces , and show partial results toward the conjecture for actions on . In particular, we prove that the spectrum associated with actions with linear part is either empty, or an interval of the form () or , whenever is an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space .
Le spectre des points fixes d’un groupe localement compact à base dénombrable est défini comme l’ensemble des tels que chaque action par isométries affines de sur admet un point fixe. Nous montrons que cet ensemble est soit vide, soit peut s’écrire sous une des formes suivantes : , pour un certain , ou , pour un certain . 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 .
Plus généralement, nous étudions les propriétés topologiques du spectre des points fixes sur pour des espaces mesurés arbitraires , et nous montrons des résultats partiels dans le sens de la conjecture pour les actions sur . En particulier, nous prouvons que le spectre associé aux actions dont la partie linéaire est est soit vide, soit un intervalle de la forme () ou , dès que est une représentation orthogonale associée à une action ergodique préservant la mesure sur un espace mesuré de mesure finie.
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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
@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/} }
TY - JOUR AU - Lavy, Omer AU - Olivier, Baptiste TI - Fixed-point spectrum for group actions by affine isometries on $L_{p}$-spaces JO - Annales de l'Institut Fourier PY - 2021 SP - 1 EP - 26 VL - 71 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3348/ DO - 10.5802/aif.3348 LA - en ID - AIF_2021__71_1_1_0 ER -
%0 Journal Article %A Lavy, Omer %A Olivier, Baptiste %T Fixed-point spectrum for group actions by affine isometries on $L_{p}$-spaces %J Annales de l'Institut Fourier %D 2021 %P 1-26 %V 71 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3348/ %R 10.5802/aif.3348 %G en %F AIF_2021__71_1_1_0
Lavy, Omer; Olivier, Baptiste. Fixed-point spectrum for group actions by affine isometries on $L_{p}$-spaces. Annales de l'Institut Fourier, Volume 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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