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

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