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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)

20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20F05 Generators, relations, and presentations of groups
Full Text: DOI arXiv
[1] 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
[2] B N Apanasov, Geometrically finite hyperbolic structures on manifolds, Ann. Global Anal. Geom. 1 (1983) 1 · Zbl 0531.57012 · doi:10.1007/BF02329729
[3] 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
[4] B H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245 · Zbl 0789.57007 · doi:10.1006/jfan.1993.1052
[5] 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
[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
[7] B H Bowditch, Relatively hyperbolic groups, Preprint, University of Southampton (1999) · Zbl 1259.20052 · eprints.soton.ac.uk
[8] S G Brick, On Dehn functions and products of groups, Trans. Amer. Math. Soc. 335 (1993) 369 · Zbl 0892.57001 · doi:10.2307/2154273
[9] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999) · Zbl 0988.53001
[10] 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
[11] 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
[12] 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
[13] 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
[14] V Efromovich, On proximity geometry of Riemannian manifolds, Amer. Math. Soc. Transl. (2) 39 (1964) 167 · Zbl 0152.39201
[15] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027 · doi:10.1007/s000390050075
[16] 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
[17] 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
[18] 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
[19] M Gromov, Hyperbolic groups (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015
[20] 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
[21] 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
[22] 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
[23] 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
[24] 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
[25] E Martínez-Pedroza, A note on quasiconvexity and relative hyperbolic structures · Zbl 1276.20055 · doi:10.1007/s10711-011-9610-3
[26] 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
[27] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) · Zbl 1093.20025
[28] 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
[29] D Y Rebbechi, Algorithmic properties of relatively hyperbolic groups, PhD thesis, Rutgers University (2001) · arxiv:math.GR/0302245
[30] H Short, Quasiconvexity and a theorem of Howson’s (editors É Ghys, A Haefliger), World Sci. Publ. (1991) 168 · Zbl 0869.20023
[31] 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
[32] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) · msri.org
[33] W P Thurston, Three-dimensional geometry and topology. Vol. 1, <span class=”textrm”>(</span>S Levy, editor), Princeton Math. Series 35, Princeton Univ. Press (1997) · Zbl 0873.57001
[34] P Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157 · Zbl 0855.30036
[35] 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
[36] 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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