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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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