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Relative hyperbolicity and relative quasiconvexity for countable groups. (English) Zbl 1202.20046
This rather useful paper has two objectives: 1) It proves the equivalence of various definitions of relative hyperbolicity. The notion was introduced by M. Gromov [in: Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] and worked out from various perspectives by Farb, Bowditch, Osin and others. 2) To define several equivalent notions of relative quasiconvexity, unifying work of Dahmani and Osin.
$$G$$ is a countable group and $$\mathbb{P}$$ a collection of subgroups. The author proves the equivalence of the following definitions of relative hyperbolicity of $$G$$ with respect to $$\mathbb{P}$$.
1) $$(G;\mathbb{P})$$ has a geometrically finite convergence group action on a compact, metrizable space $$M$$.
2) $$G$$ has a properly discontinuous action on a proper hyperbolic space $$X$$ such that the induced convergence group action on $$\partial X$$ is geometrically finite. $$\mathbb{P}$$ is a set of representatives of the conjugacy classes of maximal parabolic subgroups.
3) $$G$$ has a properly discontinuous action on a proper hyperbolic space $$X$$, and $$\mathbb{P}$$ is a set of representatives of the conjugacy classes of maximal parabolic subgroups. There is a $$G$$-equivariant collection of disjoint horoballs centered at the parabolic points of $$G$$, with union $$U$$ open in $$X$$, such that the quotient of $$X\setminus U$$ by the action of $$G$$ is compact.
4) A graph $$K$$ is fine if each edge of $$K$$ is contained in only finitely many circuits of length $$n$$ for each $$n$$. $$G$$ acts on a fine hyperbolic graph $$K$$ with finite edge stabilizers and finitely many orbits of edges. $$\mathbb{P}$$ is a set of representatives of the conjugacy classes of infinite vertex stabilizers.
5) $$G$$ is finitely generated relative to $$\mathbb{P}$$, and each $$P_i\in\mathbb{P}$$ is infinite. For some (every) finite relative generating set $$S$$, the coned-off Cayley graph is hyperbolic and $$(G,\mathbb{P},S)$$ has bounded coset penetration.
6) $$\mathbb{P}$$ is a finite collection of infinite subgroups of a countable group $$G$$. $$(G,\mathbb{P})$$ has a finite relative presentation, and the relative Dehn function is well-defined and linear for some/every finite relative presentation.
The author next gives several definitions of relative quasiconvexity and proves their equivalence.
(i) A subgroup $$H\subset G$$ is relatively quasiconvex if the following holds. Let $$M$$ be some (any) compact, metrizable space on which $$(G,\mathbb{P})$$ acts as a geometrically finite convergence group. Then the induced convergence action of $$H$$ on the limit set $$\Lambda H\subset M$$ is geometrically finite.
(ii) Let $$X$$ be some (any) proper hyperbolic space on which $$(G,\mathbb{P})$$ has a cusp uniform action. Then either $$H$$ is finite, $$H$$ is parabolic, or $$H$$ has a cusp uniform action on a geodesic hyperbolic space $$Y$$ quasi-isometric to the subspace $$\mathbf{join}(\Lambda H)\subset X$$, where $$\Lambda H$$ denotes the limit set of $$H$$.
(iii) Let $$(X,\rho)$$ be some (any) proper hyperbolic space on which $$(G,\mathbb{P})$$ has a cusp uniform action. Let $$X\setminus U$$ be some (any) truncated space for $$G$$ acting on $$X$$. For some (any) basepoint $$x\in X\setminus U$$ there is a constant $$\mu\geq 0$$ such that whenever $$c$$ is a geodesic in $$X$$ with endpoints in the orbit $$Hx$$, we have $$c\cap X\subset\mathcal N_\mu(Hx)$$ where the neighborhood is taken with respect to the metric on $$X$$.
(iv) Let $$(X,\rho)$$ be some (any) proper $$\delta$$ hyperbolic space on which $$(G,\mathbb{P})$$ has a cusp uniform action. Let $$X\setminus U$$ be some (any) truncated space for $$G$$ acting on $$X$$. Then each pair of horoballs of $$U$$ is separated by at least a distance $$r$$, where $$r$$ is a constant with a specific dependence on $$\delta$$. (See Lemma 6.8 of the paper for the nature of the dependence.)
(v) Let $$S$$ be some (any) finite relative generating set for $$(G,\mathbb{P})$$ and let $$\mathbb{P}$$ be the union of all $$P_i\in\mathbb{P}$$. Consider the Cayley graph $$\overline\Gamma$$ with generating set $$S\cup\mathbb{P}$$. Let $$d$$ be some (any) proper, left invariant metric on $$G$$. Then there is a constant $$\kappa=\kappa(S,d)$$ such that for each geodesic $$c$$ in $$\overline\Gamma$$ connecting two points of $$H$$, every vertex of $$c$$ lies within a $$d$$-distance $$\kappa$$ of $$H$$.
Reviewer: Mahan Mj (Howrah)

##### MSC:
 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 20F05 Generators, relations, and presentations of groups
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##### References:
  I Agol, D Groves, J F Manning, Residual finiteness, QCERF and fillings of hyperbolic groups, Geom. Topol. 13 (2009) 1043 · Zbl 1229.20037 · doi:10.2140/gt.2009.13.1043 · arxiv:0802.0709  B N Apanasov, Geometrically finite hyperbolic structures on manifolds, Ann. Global Anal. Geom. 1 (1983) 1 · Zbl 0531.57012 · doi:10.1007/BF02329729  A F Beardon, B Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974) 1 · Zbl 0277.30017 · doi:10.1007/BF02392106  B H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245 · Zbl 0789.57007 · doi:10.1006/jfan.1993.1052  B H Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995) 229 · Zbl 0877.57018 · doi:10.1215/S0012-7094-95-07709-6  B H Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998) 643 · Zbl 0906.20022 · doi:10.1090/S0894-0347-98-00264-1  B H Bowditch, Relatively hyperbolic groups, Preprint, University of Southampton (1999) · Zbl 1259.20052 · eprints.soton.ac.uk  S G Brick, On Dehn functions and products of groups, Trans. Amer. Math. Soc. 335 (1993) 369 · Zbl 0892.57001 · doi:10.2307/2154273  M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999) · Zbl 0988.53001  J W Cannon, D Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992) 419 · Zbl 0761.57008 · doi:10.2307/2154172  F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933 · Zbl 1037.20042 · doi:10.2140/gt.2003.7.933 · eudml:123509 · arxiv:math/0203258  F Dahmani, Les groupes relativement hyperboliques et leurs bords, Thèse, l’Université Louis Pasteur (Strasbourg I), Prépublication de l’Institut de Recherche Mathématique Avancée, 2003/13 (2003) · www-irma.u-strasbg.fr  C Drutu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959 · Zbl 1101.20025 · doi:10.1016/j.top.2005.03.003  V Efromovich, On proximity geometry of Riemannian manifolds, Amer. Math. Soc. Transl. (2) 39 (1964) 167 · Zbl 0152.39201  B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027 · doi:10.1007/s000390050075  E M Freden, Negatively curved groups have the convergence property. I, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995) 333 · Zbl 0847.20031 · emis:journals/AASF/Vol20/vol20.html · eudml:233782  H Garland, M S Raghunathan, Fundamental domains for lattices in $$({\R})$$-rank $$1$$ semisimple Lie groups, Ann. of Math. $$(2)$$ 92 (1970) 279 · Zbl 0206.03603 · doi:10.2307/1970838  F W Gehring, G J Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. $$(3)$$ 55 (1987) 331 · Zbl 0628.30027 · doi:10.1093/plms/s3-55_2.331  M Gromov, Hyperbolic groups (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015  D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317 · Zbl 1211.20038 · doi:10.1007/s11856-008-1070-6  V S Guba, M V Sapir, On Dehn functions of free products of groups, Proc. Amer. Math. Soc. 127 (1999) 1885 · Zbl 0942.20023 · doi:10.1090/S0002-9939-99-04579-7  G C Hruska, Geometric invariants of spaces with isolated flats, Topology 44 (2005) 441 · Zbl 1120.20046 · doi:10.1016/j.top.2004.10.001  A Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal. 1 (1991) 406 · Zbl 0786.22017 · doi:10.1007/BF01895641 · eudml:58130  J F Manning, E Martínez-Pedroza, Separation of relatively quasiconvex subgroups, Pacific J. Math. 244 (2010) 309 · Zbl 1201.20024 · doi:10.2140/pjm.2010.244.309 · pjm.math.berkeley.edu  E Martínez-Pedroza, A note on quasiconvexity and relative hyperbolic structures · Zbl 1276.20055 · doi:10.1007/s10711-011-9610-3  E Martínez-Pedroza, Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009) 317 · Zbl 1186.20029 · doi:10.4171/GGD/59 · www.ems-ph.org  D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) · Zbl 1093.20025  D V Osin, Peripheral fillings of relatively hyperbolic groups, Invent. Math. 167 (2007) 295 · Zbl 1116.20031 · doi:10.1007/s00222-006-0012-3 · arxiv:math/0510195  D Y Rebbechi, Algorithmic properties of relatively hyperbolic groups, PhD thesis, Rutgers University (2001) · arxiv:math.GR/0302245  H Short, Quasiconvexity and a theorem of Howson’s (editors É Ghys, A Haefliger), World Sci. Publ. (1991) 168 · Zbl 0869.20023  P Susskind, G A Swarup, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 (1992) 233 · Zbl 0791.30039 · doi:10.2307/2374703  W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) · msri.org  W P Thurston, Three-dimensional geometry and topology. Vol. 1, (S Levy, editor), Princeton Math. Series 35, Princeton Univ. Press (1997) · Zbl 0873.57001  P Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157 · Zbl 0855.30036  P Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998) 71 · Zbl 0909.30034 · doi:10.1515/crll.1998.081  A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41 · Zbl 1043.20020 · doi:10.1515/crll.2004.007
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