2020/08/21 by Rourke, Shane O
#20E08 17B30 20E22 20F65 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2008.09449
Let Λ be an ordered abelian group, Aut+(Λ) the group of order-preserving automorphisms of Λ, G a group and α:G\toAut+(Λ) a homomorphism. An α-affine action of G on a Λ-tree X is one that satisfies d(gx,gy)=αgd(x,y) (x,y∈ X, g∈ 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 UT(n,ℝ) and groups T^*(n,ℝ) of upper triangular matrices over 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 Λ0 and Λ1 and an orientation-preserving affine action of G on Λ0, we obtain another such action of the wreath product G\wr Λ1 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'.