zbMATH — the first resource for mathematics

Relative quasiconvexity using fine hyperbolic graphs. (English) Zbl 1229.20038
The authors give a new definition of relatively quasiconvex subgroups of a relatively hyperbolic group. The definition is in the context of Bowditch’s approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. The notion applies for all countable and a class of uncountable relatively hyperbolic groups. The authors show that it is equivalent to the definitions studied by Hruska for countable relatively hyperbolic groups. The authors also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic.

20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F06 Cancellation theory of groups; application of van Kampen diagrams
Full Text: DOI arXiv
[1] B Bowditch, Relatively hyperbolic groups, Preprint (1999) · Zbl 1259.20052
[2] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999) · Zbl 0988.53001
[3] F Dahmani, Les groupes relativement hyperboliques et leurs bords, PhD Thesis, Univ. Louis Pasteur (Strasbourg I) (2003); Prépublication de l’Inst. de Recherche Math. Avancée 2003/13 (2003)
[4] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027
[5] G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807 · Zbl 1202.20046
[6] J F Manning, E Martínez-Pedroza, Separation of relatively quasiconvex subgroups, Pacific J. Math. 244 (2010) 309 · Zbl 1201.20024
[7] E Martínez-Pedroza, Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009) 317 · Zbl 1186.20029
[8] E Martínez-Pedroza, D T Wise, Local quasiconvexity of groups acting on small cancellation complexes, to appear in J. Pure Appl. Algebra · Zbl 1241.20047
[9] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 no. 843, Amer. Math. Soc. (2006) · Zbl 1093.20025
[10] P Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998) 71 · Zbl 0909.30034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.