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Dehn filling in relatively hyperbolic groups. (English) Zbl 1211.20038
A finitely generated group is said to be word hyperbolic if it acts properly and co-compactly on a metric space in which every geodesic triangle is \(\delta\)-thin for some \(\delta\). Roughly speaking, a group is said to be hyperbolic relative to some collection \(\mathcal P\) of subgroups if the geometry of \(G\) is hyperbolic except for that part corresponding to \(\mathcal P\).
The authors accomplish three goals in this paper. They introduce a new space (the “cusped space”) for studying relatively hyperbolic groups; they construct a pair of useful bicombings on that space; and they extend Thurston’s Hyperbolic Dehn Surgery Theorem to the context of torsion-free relatively hyperbolic groups.
The cusped space is constructed by gluing a collection of “combinatorial horoballs” (one for each coset of each group in \(\mathcal P\)) onto the Cayley graph of \(G\). The resulting space is then hyperbolic precisely when \(G\) is hyperbolic relative to \(\mathcal P\). The space is a graph with the edge-length metric that allows the authors to adapt a number of constructions and results in word hyperbolic groups to the relative case.
Next the authors define preferred paths for the cusped space, which give rise to a bicombing that is a relatively hyperbolic version of a construction of I. Mineyev [Geom. Funct. Anal. 11, No. 4, 807-839 (2001; Zbl 1013.20034)]. The authors expect that since Mineyev’s work has many applications, their construction should allow many results to be extended to the relatively hyperbolic setting.
Finally, the authors apply the notion of preferred paths to prove a group theoretic analog of Dehn filling. A corollary of this theorem along with another result from this paper allow the authors to unify a number of known results.

MSC:
20F65 Geometric group theory
57M50 General geometric structures on low-dimensional manifolds
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
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References:
[1] I. Agol, Bounds on exceptional Dehn filling, Geometry and Topology 4 (2000), 431–449 (electronic). · Zbl 0959.57009
[2] E. Alibegović, A combination theorem for relatively hyperbolic groups, Bulletin of the London Mathematical Society 37 (2005), 459–466. · Zbl 1074.57001
[3] D. J. Allcock and S. M. Gersten, A homological characterization of hyperbolic groups, Inventiones Mathematicae 135 (1999), 723–742. · Zbl 0909.20032
[4] B. Bowditch, Relatively hyperbolic groups, preprint, available at www.maths.soton.ac.uk/staff/Bowditch/preprints.html, 1999.
[5] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319 Springer-Verlag, Berlin, 1999. · Zbl 0988.53001
[6] I. Bumagin, On definitions of relatively hyperbolic groups, in Geometric Methods in Group Theory, Contemp. Math., vol. 372, Amer. Math. Soc., Providence, RI, 2005, pp. 189–196. · Zbl 1091.20029
[7] J. W. Cannon and D. Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Transactions of the American Mathematical Society 330 (1992), 419–431. · Zbl 0761.57008
[8] F. Dahmani, Classifying spaces and boundaries for relatively hyperbolic groups, Proceedings of the London Mathematical Society (3), 86 (2003), 666–684. · Zbl 1031.20039
[9] F. Dahmani, Combination of convergence groups, Geometry and Topology 7 (2003), 933–963 (electronic). · Zbl 1037.20042
[10] F. Dahmani, Les Groupes Relativement Hyperboliqes et Leurs Bords, PhD thesis, Strasbourg, 2003.
[11] C. Druţu and M. Sapir, Relatively hyperbolic groups with rapid decay property, International Mathematics Research Notices 19 (2005), 1181–1194. · Zbl 1077.22006
[12] C. Druţu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), 959–1058. With an appendix by Denis Osin and Sapir. · Zbl 1101.20025
[13] B. Farb, Relatively hyperbolic groups, Geometric and Functional Analysis 8 (1998), 810–840. · Zbl 0985.20027
[14] K. Fujiwara and J. F. Manning, CAT(0) and CAT() fillings of hyperbolic manifolds, in preparation. · Zbl 1211.53066
[15] S. Gersten, A cohomological characterisation of hyperbolic groups, preprint, 1996.
[16] M. Gromov, Hyperbolic groups, in Essays in Group Theory, Mathematical Sciences Research Institute Publications, vol. 8. Springer, New York, 1987, pp. 75–263. · Zbl 0634.20015
[17] D. Groves, Actions of relatively hyperbolic groups on strongly bolic spaces, in preparation.
[18] D. Groves, Limit groups for relatively hyperbolic groups, I: The basic tools, preprint, arXiv: math.GR/0412492, 2004.
[19] D. Groves and J. Manning, Relative bounded cohomology and relatively hyperbolic groups, in preparation. · Zbl 1211.20038
[20] D. Groves, J. Manning, and D. Osin, in preparation.
[21] D. Groves and J. F. Manning, Fillings, finite generation and direct limits of relatively hyperbolic groups, Groups, Geometry, and Dynamics 1 (2007), 329–342. · Zbl 1169.20023
[22] C. D. Hodgson and S. Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Annals of Mathematics 162 (2005), 67–421. · Zbl 1087.57011
[23] G. Hruska, Relative hyperbolicity and relative quasi-convexity for countable groups, preprint, arXiv:0801.4596, 2008.
[24] G. C. Hruska and B. Kleiner, Hadamard spaces with isolated flats, Geometry and Topology 9 (2005), 1501–1538 (electronic). With an appendix by the authors and Mohamad Hindawi. · Zbl 1087.20034
[25] C. Hummel and V. Schroeder, Cusp closing in rank one symmetric spaces, Inventiones Mathematicae 123 (1996), 283–307. · Zbl 0860.53025
[26] M. Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140 (2000), 243–282. · Zbl 0947.57016
[27] M. Lackenby, Attaching handlebodies to 3-manifolds, Geometry and Topology 6 (2002), 889–904 (electronic). · Zbl 1021.57010
[28] V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Inventiones Mathematicae 149 (2002), 1–95. · Zbl 1084.19003
[29] I. Mineyev, Straightening and bounded cohomology of hyperbolic groups, Geometric and Functional Analysis 11 (2001), 807–839. · Zbl 1013.20034
[30] I. Mineyev, Bounded cohomology characterizes hyperbolic groups, The Quarterly Journal of Mathematics 53 (2002), 59–73. · Zbl 1013.20048
[31] I. Mineyev, Flows and joins of metric spaces, Geometry and Topology 9 (2005), 403–482 (electronic). · Zbl 1137.37314
[32] I. Mineyev, N. Monod, and Y. Shalom, Ideal bicombings for hyperbolic groups and applications, Topology 43 (2004), 1319–1344. · Zbl 1137.20033
[33] I. Mineyev and G. Yu, The Baum-Connes conjecture for hyperbolic groups, Inventiones Mathematicae 149 (2002), 97–122. · Zbl 1038.20030
[34] L. Mosher and M. Sageev, Nonmanifold hyperbolic groups of high cohomological dimension, preprint, available at http://newark.rutgers.edu/\(\sim\)mosher/ .
[35] D. V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Memoirs of the American Mathematical Society 179(843) (2006), vi+100. · Zbl 1093.20025
[36] D. V. Osin, Peripheral fillings of relatively hyperbolic groups, Inventiones Mathematicae 167 (2007), 295–326. · Zbl 1116.20031
[37] P. Papasoglu, Strongly geodesically automatic groups are hyperbolic, Inventiones Mathematicae 121 (1995), 323–334. · Zbl 0834.20040
[38] D. Rebbechi, Algorithmic Properties of Relatively Hyperbolic Groups. PhD thesis, Rutgers, Newark, 2001.
[39] V. Schroeder, A cusp closing theorem Proceedings of the American Mathematical Society 106 (1989), 797–802. · Zbl 0678.53034
[40] W. P. Thurston, Geometry and Topology of Three-Manifolds, Princeton lecture notes, Princeton, 1979.
[41] A. Yaman, A topological characterisation of relatively hyperbolic groups Journal für Reine und Angewandte Mathematik 566 (2004), 41–89. · Zbl 1043.20020
[42] G. Yu, Hyperbolic groups admit proper affine isometric actions on l p -spaces, Geometric and Functional Analysis 15 (2005), 1144–1151. · Zbl 1112.46054
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