Almost-invariant elements of group actions on Lip$_0$ spaces

TL;DR

This paper investigates conditions under which almost-invariant functionals on Lipschitz-free spaces become truly invariant under group actions, connecting the problem to properties of the acting groups and quasimorphisms.

Contribution

It introduces new conditions ensuring invariance of functionals in Lipschitz-free spaces under group actions, extending previous understanding of almost-invariance.

Findings

Certain group actions guarantee invariance of functionals

Connections established between invariance and group properties

Relations to quasimorphisms are explored

Abstract

This work is motivated by a question published in E. Glasner's paper On a question of Kazhdan and Yom Din regarding the possibility to approximate functionals on a Banach space which are almost invariant with respect to an action of a discrete group by functionals that are invariant. We study the case when the Banach space is a Lipschitz-free space equipped with an action induced by an action by isometries on the underlaying space. We find a few different conditions sufficient for the answer to be positive; for example the case of free or finitely presented groups endowed with left-invariant metrics acting on themselves by translations. Relations to quasimorphisms are also briefly studied.