Dehn fillings and elementary splittings
Authors:
Daniel Groves and Jason Fox Manning
Journal:
Trans. Amer. Math. Soc. 370 (2018), 3017-3051
MSC (2010):
Primary 20F65, 20F67, 57M50
DOI:
https://doi.org/10.1090/tran/7017
Published electronically:
January 18, 2018
MathSciNet review:
3766840
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We consider conditions on relatively hyperbolic groups about the nonexistence of certain kinds of splittings and show these properties persist in long Dehn fillings. We deduce that certain connectivity properties of the Bowditch boundary persist under long fillings.
- Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431–449. MR 1799796, DOI https://doi.org/10.2140/gt.2000.4.431
- Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553, DOI https://doi.org/10.1016/j.procs.2013.05.269
- Roger C. Alperin and Kenneth N. Moss, Complete trees for groups with a real-valued length function, J. London Math. Soc. (2) 31 (1985), no. 1, 55–68. MR 810562, DOI https://doi.org/10.1112/jlms/s2-31.1.55
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
- Mladen Bestvina, $\Bbb R$-trees in topology, geometry, and group theory, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 55–91. MR 1886668
- Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287–321. MR 1346208, DOI https://doi.org/10.1007/BF01884300
- Steven A. Bleiler and Craig D. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), no. 3, 809–833. MR 1396779, DOI https://doi.org/10.1016/0040-9383%2895%2900040-2
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
- Brian H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145–186. MR 1638764, DOI https://doi.org/10.1007/BF02392898
- B. H. Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999), no. 3, 509–527. MR 1680044, DOI https://doi.org/10.1007/PL00004703
- B. H. Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999), no. 9, 3673–3686. MR 1624089, DOI https://doi.org/10.1090/S0002-9947-99-02388-0
- B. H. Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc. 353 (2001), no. 10, 4057–4082. MR 1837220, DOI https://doi.org/10.1090/S0002-9947-01-02835-5
- B. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012), no. 3, 1250016, 66. MR 2922380, DOI https://doi.org/10.1142/S0218196712500166
- M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994
- I. M. Chiswell, Abstract length functions in groups, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 451–463. MR 427480, DOI https://doi.org/10.1017/S0305004100053093
- Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353, DOI https://doi.org/10.1007/BF01231540
- F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159, DOI https://doi.org/10.1090/memo/1156
- M. J. Dunwoody and M. E. Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999), no. 1, 25–44. MR 1664694, DOI https://doi.org/10.1007/s002220050278
- B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810–840. MR 1650094, DOI https://doi.org/10.1007/s000390050075
- David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI https://doi.org/10.2307/2946597
- Daniel Groves and Jason Fox Manning, Quasiconvexity and Dehn filling, in preparation.
- Daniel Groves and Jason Fox Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008), 317–429. MR 2448064, DOI https://doi.org/10.1007/s11856-008-1070-6
- Bradley W. Groff, Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups, J. Topol. Anal. 5 (2013), no. 4, 451–475. MR 3152211, DOI https://doi.org/10.1142/S1793525313500192
- Vincent Guirardel, Limit groups and groups acting freely on $\Bbb R^n$-trees, Geom. Topol. 8 (2004), 1427–1470. MR 2119301, DOI https://doi.org/10.2140/gt.2004.8.1427
- Vincent Guirardel, Actions of finitely generated groups on $\Bbb R$-trees, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 159–211 (English, with English and French summaries). MR 2401220
- Dan P. Guralnik, Ends of cusp-uniform groups of locally connected continua. I, Internat. J. Algebra Comput. 15 (2005), no. 4, 765–798. MR 2160578, DOI https://doi.org/10.1142/S0218196705002499
- G. Christopher Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010), no. 3, 1807–1856. MR 2684983, DOI https://doi.org/10.2140/agt.2010.10.1807
- Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282. MR 1756996, DOI https://doi.org/10.1007/s002220000047
- Denis V. Osin, Elementary subgroups of relatively hyperbolic groups and bounded generation, Internat. J. Algebra Comput. 16 (2006), no. 1, 99–118. MR 2217644, DOI https://doi.org/10.1142/S0218196706002901
- Denis V. Osin, Peripheral fillings of relatively hyperbolic groups, Invent. Math. 167 (2007), no. 2, 295–326. MR 2270456, DOI https://doi.org/10.1007/s00222-006-0012-3
- Panos Papasoglu and Eric Swenson, From continua to $\Bbb R$-trees, Algebr. Geom. Topol. 6 (2006), 1759–1784. MR 2263049, DOI https://doi.org/10.2140/agt.2006.6.1759
- Panos Papasoglu and Eric Swenson, The cactus tree of a metric space, Algebr. Geom. Topol. 11 (2011), no. 5, 2547–2578. MR 2836294, DOI https://doi.org/10.2140/agt.2011.11.2547
- E. Rips and Z. Sela, Structure and rigidity in hyperbolic groups. I, Geom. Funct. Anal. 4 (1994), no. 3, 337–371. MR 1274119, DOI https://doi.org/10.1007/BF01896245
- Cornelius Reinfeldt and Richard Weidmann, Makanin-Razborov diagrams for hyperbolic groups, Preprint, 2014.
- Viktor Schroeder, A cusp closing theorem, Proc. Amer. Math. Soc. 106 (1989), no. 3, 797–802. MR 957267, DOI https://doi.org/10.1090/S0002-9939-1989-0957267-6
- Z. Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997), no. 3, 527–565. MR 1465334, DOI https://doi.org/10.1007/s002220050172
- Zlil Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105. MR 1863735, DOI https://doi.org/10.1007/s10240-001-8188-y
- William P. Thurston, Geometry and topology of three-manifolds, Princeton lecture notes available at http://www.msri.org/publications/books/gt3m/, 1980.
- Pekka Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157–187. MR 1313451
- Asli Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004), 41–89. MR 2039323, DOI https://doi.org/10.1515/crll.2004.007
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F65, 20F67, 57M50
Retrieve articles in all journals with MSC (2010): 20F65, 20F67, 57M50
Additional Information
Daniel Groves
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
MR Author ID:
642547
Email:
groves@math.uic.edu
Jason Fox Manning
Affiliation:
Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853
MR Author ID:
690777
Email:
jfmanning@math.cornell.edu
Received by editor(s):
March 31, 2016
Received by editor(s) in revised form:
July 6, 2016
Published electronically:
January 18, 2018
Additional Notes:
The results in this paper were instigated at the Mathematisches Forschungsinstitut Oberwolfach in June 2011. Both authors were supported in part by the NSF (under grants DMS-0953794 and DMS-1462263)
Article copyright:
© Copyright 2018
American Mathematical Society