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Elementary subgroups of relatively hyperbolic groups and bounded generation. (English) Zbl 1100.20033
Originally, the notion of a relatively hyperbolic group was proposed by M. Gromov [see Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] in order to generalize various examples of algebraic and geometric nature of fundamental groups of finite-volume non-compact Riemannian manifolds of pinched negative curvature. Gromov’s idea has been elaborated by B. Bowditch in his preprint ‘Relatively hyperbolic groups’ [Univ. Southampton (1998)]. In 1994, B. Farb, in his PhD Univ. Princeton thesis [published as article in Geom. Funct. Anal. 8, No. 5, 810-840 (1998; Zbl 0985.20027)], proposed an alternative approach.
The paper under review continues the investigation initiated in the author’s article [in Mem. Am. Math. Soc. 843 (2006; Zbl 1093.20025)]. The author mentions that it is the second article in a sequence of three and is supposed to establish the background for the article entitled ‘Relatively hyperbolic groups and embedding theorems’ [available at arxiv: math. GR/0411039], where he uses relative hyperbolicity to prove certain embedding theorems for countable groups.
Definition. Let $$G$$ be a group hyperbolic relative to a collection of subgroups $$\{H_\lambda$$, $$\lambda \in\Lambda\}$$. A subgroup $$Q\subseteq G$$ is said to be hyperbolically embedded into $$G$$, if $$G$$ is hyperbolic relative to $$\{H_\lambda,\;\lambda\in\Lambda\}\cup\{Q\}$$.
For every element $$g\in G$$, we denote by $$|g|_S$$ its relative length, that is the word length with respect to the generating set $$S$$. Let $${\mathcal H}=\bigcup_{\lambda\in\Lambda}(H_\lambda\setminus\{1\})$$.
The main result of this paper is the following Theorem. Suppose that $$G$$ is a group hyperbolic relative to a collection of subgroups $$\{H_\lambda,\;\lambda\in\Lambda\}$$ and $$Q$$ is a subgroup of $$G$$. Then $$Q$$ is hyperbolically embedded into $$G$$ if and only if the following conditions hold: (Q1) $$Q$$ is generated by a finite set $$Y$$. (Q2) There exist $$\nu,c\geq 0$$ such that for any element $$q\in Q$$, we have $$|q|_Y\geq\nu|q|_S +c$$. (Q3) For any $$g\in G\setminus Q$$, we have $$(Q\cap Q^g)<\infty$$.
There are proved two corollaries: 1. If $$Q$$ is hyperbolically embedded into $$G$$, then $$Q$$ is a hyperbolic group. 2. For any hyperbolic element $$g\in G$$ of infinite order, $$E(g)$$ is hyperbolic embedded into $$G$$. ($$E(g)$$ is an elementary subgroup of $$G$$ and $$g\in E(g)$$.)

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 20F05 Generators, relations, and presentations of groups 20F69 Asymptotic properties of groups 20E07 Subgroup theorems; subgroup growth
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