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 ${\ell }^1$.

17 pages, 1 figure (created with GeoGebra). Comments welcome