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

TL;DR

This paper establishes that the Lipschitz free space of countable simplicial quasi-trees is isomorphic to ^1, and demonstrates that certain groups, including those with Property (QT) and 3-manifold groups, act properly on ^1 via Lipschitz affine actions.

Contribution

It proves the isomorphism between Lipschitz free spaces of quasi-trees and ^1, and constructs proper affine actions on ^1 for specific groups.

Findings

Lipschitz free space of countable simplicial quasi-trees is isomorphic to ^1.

Groups with Property (QT) admit proper affine actions on ^1.

3-manifold groups have proper Lipschitz affine actions on ^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 $\ell^1$.