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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)].

##### 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
##### Citations:
Zbl 0634.20015; Zbl 0985.20027; Zbl 0834.20038; Zbl 0845.57002
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##### References:
 [1] B. H. Bowditch,Geometrical finiteness with variable negative curvature, Duke Mathematical Journal77 (1995), 229–274. · Zbl 0877.57018 [2] B. H. Bowditch,Relatively hyperbolic groups, preprint, Southampton (1999). · Zbl 0952.20032 [3] M. Coornaert, T. Delzant and A. Papadopoulos,Géométrie et théorie des groupes; les groupes hyperboliques de Gromov, Lecture Notes in Mathematics1441, Springer, Berlin, 1990. · Zbl 0727.20018 [4] F. Dahmani,Combination of convergence groups, Geometry & Toplogy7 (2003), 933–963. · Zbl 1037.20042 [5] F. Dahmani and A. Yaman,Symbolic dynamics and relatively hyperbolic groups, preprint (2002). · Zbl 1169.20022 [6] T. Delzant,L’image d’un groupe dans un groupe hyperbolique, Commentarii Mathematici Helvetici70 (1995), 267–284. · Zbl 0834.20038 [7] B. Farb,Relatively hyperbolic groups, Geometric and Functional Analysis8 (1998), 810–840. · Zbl 0985.20027 [8] E. Ghys and P. de la Harpe,Sur les groupes hyperboliques d’après Mikhael Gromov, Swiss seminar, Birkhäuser, Basel, 1990. · Zbl 0731.20025 [9] M. Gromov,Hyperbolic groups, inEssays in Group Theory (S. Gersten, ed.), Mathematical Sciences Research Institute Publications, Vol. 4, Springer, New York, 1987, pp. 75–263. · Zbl 0634.20015 [10] H. Masur and Y. Minsky,Geometry of the complex of curves. I. Hyperbolicity, Inventiones Mathematicae138 (1999), 103–149. · Zbl 0941.32012 [11] C. McMullen,From dynamics on surfaces to rational points on curves, Bulletin of the American Mathematical Society37 (2000), 119–140. · Zbl 1012.11049 [12] E. Rips and Z. Sela,Canonical representatives and equations in hyperbolic groups, Inventiones Mathematicae120 (1995), 489–512. · Zbl 0845.57002 [13] W. Thurston,The Geometry and Topology of 3-Manifolds, Princeton University Press, 1978. · Zbl 0399.73039
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