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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 \(g\in E(g)\).)

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
Full Text: DOI
[1] DOI: 10.1142/S0218196792000220 · Zbl 0794.20061
[2] DOI: 10.1215/S0012-7094-95-07709-6 · Zbl 0877.57018
[3] DOI: 10.1007/978-3-662-12494-9
[4] DOI: 10.2307/2374319 · Zbl 0525.20029
[5] DOI: 10.1016/S0764-4442(01)02061-4 · Zbl 1010.20033
[6] Farb B., GAFA 8 pp 810–
[7] DOI: 10.1112/jlms/54.2.261 · Zbl 0861.20033
[8] DOI: 10.1007/978-1-4684-9167-8
[9] DOI: 10.1007/978-1-4613-9586-7_3
[10] DOI: 10.1090/S0002-9947-96-01510-3 · Zbl 0876.20023
[11] M. V. Kumar, Number Theory, Halifax, NS, 1994, CMS Conference Proceedings 15 (Amer. Math. Soc., Providence, RI, 1995) pp. 249–261.
[12] DOI: 10.1007/978-3-0348-8965-0
[13] DOI: 10.1007/978-3-642-61896-3
[14] Olshanskii A. Yu., Mat. Sbornik 182 pp 543–
[15] DOI: 10.1142/S0218196793000251 · Zbl 0830.20053
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[17] Rapinchuk A. S., Dokl. Akad. Nauk SSSR 314 pp 1327–
[18] DOI: 10.1007/BF02698832 · Zbl 0980.22017
[19] Tavgen’ I. O., Izv. Akad. Nauk SSSR Ser. Mat. 54 pp 97–
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