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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.

MSC:
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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