Fixed-point spectrum for group actions by affine isometries on L p -spaces

Abstract

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Isometric actions on Lp-spaces: dependence on the value of p

“…In particular, if is a locally compact property (T) group, for every . This was already known when is discrete [LO21], or admits a finite Kazhdan set [CK20].…”

“…The fact that these questions have a positive answer is sometimes referred to as Druţu's conjecture [Now15, LO21].…”

“…When is a second countable locally compact group, it is known that if and only if has property (T) [BFGM07], in which case we must have (in fact, by an argument of Fisher and Margulis from [FM05] that appears in [BFGM07], one can even show that and that admits a continuous action by isometries on an space without fixed points for , see also [LO21, DK18, Sal19]). Similarly, it is known that if and only if does not have the Haagerup property, in which case (see [Now06]).…”

“…Thus, Corollary 2 answers positively Question 1.1. A partial answer for spaces had already been obtained independently by Czuroń [Czu17] and Lavy and Olivier [LO21]. In [LO21] actions coming from ergodic probability measure-preserving actions were also covered.…”

“…A partial answer for spaces had already been obtained independently by Czuroń [Czu17] and Lavy and Olivier [LO21]. In [LO21] actions coming from ergodic probability measure-preserving actions were also covered. Unlike these previous results, it is worth mentioning that in Theorem 1, the linear part of the action that we construct is very different from the linear part of the original action .…”