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Height in splittings of hyperbolic groups. (English) Zbl 1059.20040
Let $$H$$ be a hyperbolic subgroup of the hyperbolic group $$G$$. Call the elements $$\{g_i\mid i=1,\dots,n\}$$ essentially distinct if $$Hg_i\neq Hg_j$$ whenever $$i\neq j$$. Conjugates of $$H$$ by essentially distinct elements are called essentially distinct conjugates. The height of an infinite subgroup $$H$$ in $$G$$ is $$n$$ if there is a collection of $$n$$ essentially distinct conjugates of $$H$$ such that the intersection of the collection is infinite and $$n$$ is maximal with respect to this property. Finite groups have height $$0$$ by definition.
The main result of R. Gitik, M. Mitra, E. Rips, and M. Sageev [Trans. Am. Math. Soc. 350, No. 1, 321-329 (1998; Zbl 0897.20030)] is the following: if $$H$$ is a quasiconvex subgroup of a hyperbolic group $$G$$, then $$H$$ has finite height. The paper under review proves the converse of that theorem for hyperbolic groups $$G$$ which split as $$G=G_1*_HG_2$$ or $$G=G_1*_H$$ with hyperbolic edge and vertex groups where the inclusions are quasi-isometric embeddings, giving an affirmative answer to a question of Swarup.
The main result states: Let $$G$$ be a hyperbolic group splitting over $$H$$ with hyperbolic edge and vertex groups; assume the inclusions of $$H$$ are quasi-isometric embeddings; then $$H$$ is of finite height in $$G$$ if and only if it is quasiconvex in $$G$$.
The paper includes details regarding consequences and questions related to the main theorem including malnormality, more general graphs of hyperbolic groups as well as more geometric applications.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 57M05 Fundamental group, presentations, free differential calculus
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