Accidental parabolics and relatively hyperbolic groups. (English) Zbl 1174.20014
Originally, the notion of a relatively hyperbolic group was proposed by M. Gromov [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. H. 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.
We need the following definition to formulate the main result of the paper under review. We say that a morphism from a group in a relatively hyperbolic group \(h\colon G\to\Gamma\) has an accidental parabolic if either \(h(G)\) is finite or parabolic in \(\Gamma\), or if \(h\) can be factorized through a non-trivial amalgamated free product \(f\colon G\to A*_CB\) or HNN extension \(F\colon G\to A*_C\) where \(f\) and \(F\) are surjective, and the image of \(C\) is either finite or parabolic in \(\Gamma\).
The main result of the paper is the following: Let \(G\) be a finitely presented group and \(\Gamma\) a relatively hyperbolic group. Then there are finitely many subgroups of \(\Gamma\), up to conjugacy, that are images of \(G\) in \(\Gamma\) by a morphism without accidental parabolic. In the proof are used the ideas of T. Delzant [Comment. Math. Helv. 70, No. 2, 267-284 (1995; Zbl 0834.20038)] and E. Rips and Z. Sela [Invent. Math. 120, No. 3, 489-512 (1995; Zbl 0845.57002)].
We need the following definition to formulate the main result of the paper under review. We say that a morphism from a group in a relatively hyperbolic group \(h\colon G\to\Gamma\) has an accidental parabolic if either \(h(G)\) is finite or parabolic in \(\Gamma\), or if \(h\) can be factorized through a non-trivial amalgamated free product \(f\colon G\to A*_CB\) or HNN extension \(F\colon G\to A*_C\) where \(f\) and \(F\) are surjective, and the image of \(C\) is either finite or parabolic in \(\Gamma\).
The main result of the paper is the following: Let \(G\) be a finitely presented group and \(\Gamma\) a relatively hyperbolic group. Then there are finitely many subgroups of \(\Gamma\), up to conjugacy, that are images of \(G\) in \(\Gamma\) by a morphism without accidental parabolic. In the proof are used the ideas of T. Delzant [Comment. Math. Helv. 70, No. 2, 267-284 (1995; Zbl 0834.20038)] and E. Rips and Z. Sela [Invent. Math. 120, No. 3, 489-512 (1995; Zbl 0845.57002)].
Reviewer: Andrzej Szczepański (Gdańsk)
MSC:
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 57M07 | Topological methods in group theory |
| 20E36 | Automorphisms of infinite groups |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20E07 | Subgroup theorems; subgroup growth |
| 20F65 | Geometric group theory |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
relatively hyperbolic groups; finitely presented groups; morphisms without accidental parabolics; graphs of groups; amalgamated free products; HNN extensionsReferences:
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