Geodesic
Presenting parabolic subgroups
Dahmani, François1 ; Guirardel, Vincent2
1Institut Fourier, Université de Grenoble 1, 100 rue des Maths, BP 76, 38402 St. Martin d’Hères Cedex, France
2Institut de recherche en mathématiques de Rennes, Université de Rennes 1, 263 avenue du Général Leclerc, CS 74205, 35042 Rennes Cedex, France
Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3203-3222
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Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem and generating sets of the parabolic subgroups. We also give an algorithm that finds parabolic subgroups in a given recursively enumerable class of groups.

DOI : 10.2140/agt.2013.13.3203
Classification : 20F67, 20F10
Keywords: relatively hyperbolic groups, finite presentations, van Kampen diagrams, decision problems

Dahmani, François  1   ; Guirardel, Vincent  2

1 Institut Fourier, Université de Grenoble 1, 100 rue des Maths, BP 76, 38402 St. Martin d’Hères Cedex, France
2 Institut de recherche en mathématiques de Rennes, Université de Rennes 1, 263 avenue du Général Leclerc, CS 74205, 35042 Rennes Cedex, France 
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Dahmani, François; Guirardel, Vincent. Presenting parabolic subgroups. Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3203-3222. doi: 10.2140/agt.2013.13.3203

[1] B H Bowditch, Notes on Gromov's hyperbolicity criterion for path-metric spaces, from: "Group theory from a geometrical viewpoint" (editors É Ghys, A Haefliger, A Verjovsky), World Sci. Publ., River Edge, NJ (1991) 64

[2] B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012)

[3] F Dahmani, Finding relative hyperbolic structures, Bull. Lond. Math. Soc. 40 (2008) 395

[4] V Gerasimov, L Potyagailo, Quasiconvexity in the relatively hyperbolic groups,

[5] M Gromov, Hyperbolic groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[6] D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317

[7] G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807

[8] R C Lyndon, P E Schupp, Combinatorial group theory, Classics in Mathematics 14, Springer (2001) 339

[9] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) 100

[10] P Papasoglu, An algorithm detecting hyperbolicity, from: "Geometric and computational perspectives on infinite groups" (editors G Baumslag, D Epstein, R Gilman, H Short, C Sims), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 25, Amer. Math. Soc. (1996) 193

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