Affine actions on non-archimedean trees

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 $\mathbb{R}$-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.

27 pages. Section 1.4 expanded, typos corrected from previous version

References in corpus (5)

• Limit groups and groups acting freely on R^n-trees
• Infinite words and universal free actions
• Groups with free regular length functions in Z^n
• Membership Problem in groups acting freely on Z^n-trees
• Groups acting freely on $\Lambda$-trees

Cited by in corpus (1)

• A tree-free group that is not orderable