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Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups. (English) Zbl 1138.20024
The paper under review is related to properties of finitely generated groups. In the opinion of the reviewer the most interesting are the following theorems. 1. Let $$G$$ be a strongly relatively hyperbolic group. Suppose that the peripheral subgroups of $$G$$ do not contain free non-Abelian subgroups. Let $$\Lambda$$ be a finitely generated subgroup of $$G$$ which is neither virtually cyclic nor parabolic. Assume moreover that $$\Lambda$$ does not split over a parabolic subgroup nor over a virtually cyclic subgroup. Then there are finitely many conjugacy classes in $$G$$ of injective homomorphisms $$\Lambda\to G$$ whose image is not parabolic.
2. Suppose that the peripheral subgroups of $$G$$ are not relatively hyperbolic with respect to proper subgroups. If $$\text{Out}(G)$$ is infinite then one of the following cases occurs: (1) $$G$$ splits over a virtually cyclic subgroup; (2) $$G$$ splits over a parabolic (finitely of uniformly bounded size)-by-Abelian-by-(virtually cyclic) subgroup; (3) $$G$$ can be represented as a non-trivial amalgamated product or HNN extension with one of the vertex groups a maximal parabolic subgroup of $$G$$. – Moreover, some results about co-Hopfian property of relatively hyperbolic groups are also proved.
The authors obtain the above theorems from a more general theory of groups acting on so called tree-graded spaces. They develop it from the theory of groups acting on $$\mathbb{R}$$-trees.

MSC:
 20E08 Groups acting on trees 20F67 Hyperbolic groups and nonpositively curved groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07 Subgroup theorems; subgroup growth 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 53A35 Non-Euclidean differential geometry 20F65 Geometric group theory 20E36 Automorphisms of infinite groups 57M05 Fundamental group, presentations, free differential calculus
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