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A combination theorem for strong relative hyperbolicity. (English) Zbl 1192.20027
From the introduction: We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn’s Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.
In [J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029)], M. Bestvina and M. Feighn proved a combination theorem for hyperbolic groups. Motivated by this, Swarup asked the analogous question [in Geometric group theory problem list, available at http://math.utah.edu/~bestvina], for relatively hyperbolic groups. F. Dahmani [Geom. Topol. 7, 933-963 (2003; Zbl 1037.20042)] and E. Alibegović [Bull. Lond. Math. Soc. 37, No. 3, 459-466 (2005; Zbl 1074.57001)] have proven combination theorems motivated by applications to convergence groups and limit groups.
In this paper, we prove a geometric combination theorem (as opposed to a dynamical one) for trees of (strong) relatively hyperbolic metric spaces. We use Bestvina and Feighn’s Combination Theorem [op. cit.] directly in deducing the relevant combination theorem. The conditions we impose are quite different from those of Dahmani [op. cit.] and Alibegović [op. cit.]. Our main Theorems 4.5 and 4.7 are stated below:
Strong Combination Theorem and converse: Theorem 4.5 and Theorem 4.7: Let $$X$$ be a tree $$(T)$$ of strongly relatively hyperbolic spaces satisfying: (1) the q(uasi)-i(sometrically)-embedded condition, (2) the strictly type-preserving condition, (3) the qi-preserving electrocution condition, (4) the induced tree of coned-off spaces satisfies the hallways flare condition, (5) the cone-bounded hallways strictly flare condition. Then $$X$$ is strongly hyperbolic relative to the family $$\mathcal C$$ of maximal cone-subtrees of horosphere-like spaces.
Conversely, if $$X$$ is a tree $$(T)$$ of strongly relatively hyperbolic spaces satisfying conditions (1)-(3) such that $$X$$ is strongly hyperbolic relative to the family $$\mathcal C$$ of maximal cone-subtrees of horosphere-like spaces, then the tree of spaces satisfies conditions (4)-(5).
Reviewer: Reviewer (Berlin)

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 57M50 General geometric structures on low-dimensional manifolds 20F65 Geometric group theory 20M07 Varieties and pseudovarieties of semigroups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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