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Dendrology of groups in low \({\mathbb{Q}}\)-ranks. (English) Zbl 0732.20011
The work of Bass and Serre [J. P. Serre, “Arbres, amalgames, \(SL_ 2'' \), Astérisque 46 (1977; Zbl 0369.20013)] gave much insight into the structure of groups acting on simplicial trees. It gave a topological way of looking at free products with amalgamations. In the theory of linear algebraic groups, simplicial trees arise as Bruhat-Tits buildings of rank one algebraic groups, such as \(SL_ 2\) over discretely valued fields.
The authors study actions of groups (by isometries) on \(\Lambda\)-trees, where \(\Lambda\) is an ordered abelian group. Their results hold for the case that \(\Lambda\) is a subgroup of \({\mathbb{R}}\) of \({\mathbb{Q}}\)- rank\(=\dim_{{\mathbb{Q}}}\Lambda \otimes_{{\mathbb{Z}}}{\mathbb{Q}}\) at most two. They show that if a group \(\Gamma\) acts freely and without inversions on a \(\Lambda\)-tree then \(\Gamma\) is a free product of infinite cyclic groups and surface groups. Their theorem B says that every action of a surface group \(\pi_ 1(\Sigma)\) on a \(\Lambda\)-tree satisfying some natural hypotheses has an \({\mathbb{R}}\)-completion which is the action of \(\pi_ 1(\Sigma)\) on the dual tree of a measured foliation on \(\Sigma\), in the sense of Thurston’s theory of measured foliations. Their theorems C and D give conditions under which a group \(\Gamma\) acting on a \(\Lambda\)-tree splits over certain subgroups. The results are derived from a general structure theorem involving concrete geometric actions on - what the authors call - measured foliations on singular surfaces which generalize both simplicial actions and the actions defined by measured foliations on surfaces.

20E08 Groups acting on trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M60 Group actions on manifolds and cell complexes in low dimensions
20F65 Geometric group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20G15 Linear algebraic groups over arbitrary fields
57M50 General geometric structures on low-dimensional manifolds
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