Quasi-trees, Lipschitz free spaces, and actions on $\ell^1$

Abstract

We show that the Lipschitz free space of a countable simplicial quasi-tree is isomorphic to $\ell^1$. As a consequence, every finitely generated group with Property (QT) of Bestvina--Bromberg--Fujiwara has a proper uniformly Lipschitz affine action on $\ell^1$ with quasi-isometrically embedded orbits. We also show that $3$-manifold groups admit proper uniformly Lipschitz affine actions on $\ell^1$.