Almost-invariant elements of group actions on Lip0 spaces

This work is motivated by a question published in E. Glasner’s 2022 paper, On a question of Kazhdan and Yom Din, regarding the possibility of approximating 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 underlying 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.