Affine actions on non-archimedean trees

Details Affine actions on non-archimedean trees Affine actions on non-archimedean trees

2011-12-20

arxiv.org/abs/1112.4832v2

Mathematics

Group Theory

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

Scientific paper

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.

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