A combination theorem for affine tree-free groups

2015/03/12 by Rourke, Shane O
#20E08 #20F65 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1503.03788

Abstract

Let Λ₀ be an ordered abelian group. We show how an ATF(ℤ×Λ₀) group -- that is, a group admitting a free affine action without inversions on a ℤ×Λ₀-tree -- admits a natural graph of groups decomposition, where vertex groups inherit actions on Λ₀-trees. Using recent work of various authors, it follows that a finitely generated group admitting a free affine action on a ℤⁿ-tree where no line has its orientation reversed is relatively hyperbolic with nilpotent parabolics, is locally quasiconvex, and has solvable word, conjugacy and isomorphism problems. Conversely, given a graph of groups satisfying certain conditions, we show how an affine action of its fundamental group can be constructed. Specialising to the case of free affine actions, we obtain a large class of ATF(ℤ×Λ₀) groups that do not act freely by isometries on any Λ₁-tree. We also give an example of a group that admits a free isometric action on a ℤ×ℤ-tree but which is not residually nilpotent.

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