## 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

Zbl 0369.20013
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