Affine structures, wreath products and free affine actions on linear non-archimedean trees

Abstract

Let $Λ$ be an ordered abelian group, $\mathrm{Aut}^+(Λ)$ the group of order-preserving automorphisms of $Λ$, $G$ a group and $α:G\to\mathrm{Aut}^+(Λ)$ a homomorphism. An $α$-affine action of $G$ on a $Λ$-tree $X$ is one that satisfies $d(gx,gy)=α_gd(x,y)$ ($x,y\in X$, $g\in G$). We consider classes of groups that admit a free, rigid, affine action in the case where $X=Λ$. Such groups form a much larger class than in the isometric case. We show in particular that unitriangular groups $\mathrm{UT}(n,\mathbb{R})$ and groups $T^*(n,\mathbb{R})$ of upper triangular matrices over $\mathbb{R}$ with positive diagonal entries admit free affine actions. Our proofs involve left symmetric structures on the respective Lie algebras and the associated affine structures on the groups in question. We also show that given ordered abelian groups $Λ_0$ and $Λ_1$ and an orientation-preserving affine action of $G$ on $Λ_0$, we obtain another such action of the wreath product $G\wr Λ_1$ on a suitable $Λ'$. It follows that all free soluble groups, residually free groups and locally residually torsion-free nilpotent groups admit essentially free affine actions on some $Λ'$.