Affine actions on non-archimedean trees

TL;DR

This paper explores affine group actions on $\Lambda$-trees, generalizing isometric actions and introducing new classes of groups with such actions, including Baumslag-Solitar and Heisenberg groups.

Contribution

It extends the theory of group actions on trees to affine actions on $\Lambda$-trees for general ordered abelian groups, unifying previous concepts and introducing new group classes.

Findings

Introduces affine actions by dilations on $\Lambda$-trees.

Generalizes duality between length functions and actions.

Identifies groups admitting free affine actions, including Baumslag-Solitar and Heisenberg groups.

Abstract

We initiate the study of affine actions of groups on $\Lambda$-trees for a general ordered abelian group $\Lambda$; these are actions by dilations rather than isometries. This gives a common generalisation of isometric action on a $\Lambda$-tree, and affine action on an $\R$-tree as studied by I. Liousse. The duality between based length functions and actions on $\Lambda$-trees is generalised to this setting. We are led to consider a new class of groups: those that admit a free affine action on a $\Lambda$-tree for some $\Lambda$. Examples of such groups are presented, including soluble Baumslag-Solitar groups and the discrete Heisenberg group.