Electronic Research Announcements

Author(s): D. V. Osin
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 44-52.
MSC (2000): Primary 20F65, 20F67, 57M27
Posted: April 28, 2006
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Abstract: A group-theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group

we define a peripheral filling procedure, which produces quotients of

by imitating the effect of the Dehn filling of a complete finite-volume hyperbolic 3-manifold

on the fundamental group $\pi_1(M)$ . The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of

``almost'' have the Congruence Extension Property and the group

is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings.

1.: J. Alonso, M. Bridson, Semihyperbolic groups, Proc. London Math. Soc. (3) 70 (1995), no. 1, 56-114. MR 1300841 (95j:20033)
2.: M. T. Anderson, Dehn filling and Einstein metrics in higher dimensions, Preprint, 2003; available at http://www.arxiv.org/abs/math.DG/0303260.
3.: W. J. Blok, D. Pigozzi, On the congruence extension property, Algebra Universalis 38 (1997), no. 4, 391-394. MR 1626343 (99m:08007)
4.: B. H. Bowditch, Relatively hyperbolic groups, Preprint, 1999.
5.: C. Drutu, M. Sapir, Tree-graded spaces and asymptotic cones of groups. With an appendix by D. Osin and M. Sapir, Topology 44 (2005), no. 5, 959-1058. MR 2153979 (2006d:20078)
6.: P. Eberlein, Lattices in spaces of nonpositive curvature, Annals of Math. 111 (1980), 435-476. MR 0577132 (82m:53040)
7.: D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694 (93i:20036)
8.: B. Farb, Relatively hyperbolic groups, GAFA 8 (1998), 810-840. MR 1650094 (99j:20043)
9.: M. Gromov, Hyperbolic groups, Essays in Group Theory, MSRI Series, Vol. 8 (S. M. Gersten, ed.), Springer, 1987, pp. 75-263. MR 0919829 (89e:20070)
10.: D. Groves, Limits of (certain) CAT(0) groups, II: The Hopf property and the shortening argument, Preprint, 2004; http://www.arxiv.org/abs/math.GR/0408080.
11.: D. Groves, Limits of (certain) CAT(0) groups, I: Compactification, Preprint, 2004; available at http://www.arxiv.org/abs/math.GR/0404440.
12.: D. Groves and J. Manning, Dehn filling in relatively hyperbolic groups, arXiv: math.GR/0601311.
13.: V. Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$ -trees, Geometry & Topology 8 (2004), 1427-1470. MR 2119301 (2005m:20060)
14.: Ph. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969. MR 0283083 (44:316)
15.: G. Higman, B. H. Neumann, H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247-254. MR 0032641 (11:322d)
16.: S. V. Ivanov, A. Yu. Olshanskii, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091-2138. MR 1327257 (96m:20057)
17.: A. Karrass, W. Magnus, D. Solitar, Elements of finite order in groups with a single defining relation, Comm. Pure Appl. Math. 13 (1960), 57-66. MR 0124384 (23:A1696)
18.: O. Kharlampovich, A. Myasnikov, Description of fully residually free groups and irreducible affine varieties over a free group. Summer School in Group Theory in Banff, 1996, 71-80, CRM Proc. Lecture Notes, vol. 17, Amer. Math. Soc., Providence, RI, 1999. MR 1653685 (99j:20032)
19.: R. C. Lyndon, P. E. Shupp, Combinatorial Group Theory, Springer-Verlag, 1977. MR 0577064 (58:28182)
20.: A. Yu. Olshanskii, Geometry of defining relations in groups. Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1191619 (93g:20071)
21.: A. Yu. Olshanskii, Periodic quotients of hyperbolic groups, Math. USSR Sbornik 72 (1992), no. 2, 519-541. MR 1119008 (92d:20050)
22.: A. Yu. Olshanskii, ${\rm SQ}$ -universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119-132; English transl., Sb. Math. 186 (1995), no. 8, 1199-1211. MR 1357360 (97b:20057)
23.: A. Yu. Olshanskii, M. V. Sapir, Nonamenable finitely presented torsion-by-cyclic groups, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 43-169. MR 1985031 (2004f:20061)
24.: D. V. Osin, Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems, Memoirs Amer. Math. Soc., to appear; available at http://www.arxiv.org/abs/math.GR/0404040.
25.: D. V. Osin, Elementary subgroups of hyperbolic groups and bounded generation, Int. J. Alg. Comp., to appear; available at http://www.arxiv.org/abs/math.GR/0404118.
26.: D. V. Osin, Asymptotic dimension of relatively hyperbolic groups, Internat. Math. Res. Notices 2005, no. 35, 2143-2162. MR 2181790
27.: D. Osin, Peripheral fillings of relatively hyperbolic groups, arXiv: math.GR/0510195.
28.: N. Ozawa, Boundary amenability of relatively hyperbolic groups, Preprint, 2005; available at http://www.arxiv.org/abs/math.GR/0501555.
29.: Y. D. Rebbechi, Algorithmic properties of relatively hyperbolic groups, PhD thesis, Rutgers Univ. (Newark); available at http://www.arxiv.org/abs/math.GR/0302245.
30.: D. Rolfsen, Knots and links, Math. Lect. Series 7, Publish or Perish Inc., Houston, Texas, 1976. MR 0515288 (58:24236)
31.: Z. Sela, Diophantine geometry over groups I: Makanin-Razborov diagrams, IHES Publ. Math. 93 (2001), 31-105. MR 1863735 (2002h:20061)
32.: J. Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs, vol. 4, Yale University Press, New Haven, Conn.-London, 1971. MR 0415622 (54:3705)
33.: X. Tang, Semigroups with the congruence extension property, Semigroup Forum 56 (1998), no. 2, 228-264. MR 1490295 (99i:20079)
34.: W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 0648524 (83h:57019)
35.: P. Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157-187. MR 1313451 (96c:30042)
36.: D. Wise, The residual finiteness of negatively curved polygons of finite groups, Invent. Math. 149 (2002), no. 3, 579-617. MR 1923477 (2003e:20033)
37.: M. Wu, The fuzzy congruence extension property in groups, JP J. Algebra Number Theory Appl. 2 (2002), no. 2, 153-160. MR 1942382 (2003j:20106)
38.: A. Yaman, A topological characterization of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004), 41-89. MR 2039323 (2005e:20064)
39.: G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), no. 2, 325-355. MR 1626745 (99k:57072)

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D. V. Osin
Affiliation: Department of Mathematics, City College of CUNY, New York, NY 10031
Email: denis.osin@gmail.com

DOI: 10.1090/S1079-6762-06-00159-4
PII: S 1079-6762(06)00159-4
Received by editor(s): November 7, 2005
Posted: April 28, 2006
Communicated by: Efim Zelmanov
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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