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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hyperbolic groups and free constructions

Authors: O. Kharlampovich and A. Myasnikov
Journal: Trans. Amer. Math. Soc. 350 (1998), 571-613
MSC (1991): Primary 20F06, 20E06
MathSciNet review: 1390041
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Abstract: It is proved that the property of a group to be hyperbolic is preserved under HHN-extensions and amalgamated free products provided the associated (amalgamated) subgroups satisfy certain conditions. Some more general results about the preservation of hyperbolicity under graph products are also obtained. Using these results we describe the $\mathbf{Q}$-completion $(\mathbf{Q}$ is the field of rationals) $G^{\mathbf{Q}}$ of a torsion-free hyperbolic group $G$ as a union of an effective chain of hyperbolic subgroups, and solve the conjugacy problem in $G^{\mathbf{Q}}$.

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  • 1. G Baumslag. On free $ \mathbf{D}$-groups, Comm. Pure Appl. Math. 18 (1965), 25-30. MR 31:1293
  • 2. -, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217-303. MR 23:A191
  • 3. G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991), 229-316. MR 93a:20048
  • 4. -, Automatic groups and amalgams-a survey, In Algorithms and Classification in Combinatorial Group Theory, Math. Sci. Res. Inst. Publ., vol. 23, Springer-Verlag, Berlin, 1992, pp. 179-194. MR 94g:20040
  • 5. M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Diff. Geom. 35 (1992), 85-101. MR 93d:53053
  • 6. S. M. Gersten and H. B. Short, Rational subgroups of biatomatic groups, Ann. of Math. (2) 134 (1991), 125-158. MR 92g:20092
  • 7. R. Gitik, On combination theorems for negatively curved groups, Internat. J. Algebra Comput. 6 (1996), 751-760. CMP 97:04
  • 8. M. Gromov, Hyperbolic groups, Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, Berlin, 1987, pp. 75-263. MR 89e:20070
  • 9. I. Kapovich, On a theorem of G. Baumslag, Proc. Special Session Combinatorial Group Theory and Related Topics (Brooklyn, NY, 1994), Amer. Math. Soc., Providence, RI (to appear).
  • 10. W. Magnus, A. Karras, and D. Solitar, Combinatorial group theory, Interscience, New York, 1966. MR 34:7617
  • 11. K. V. Mikhajlovskii and A. Yu. Ol'shanskii, Some constructions relating to hyperbolic groups, 1994, Proc. Int. Conf. on Cohomological and Geometric Methods in Group Theory (to appear).
  • 12. A. G. Myasnikov and V. N. Remeslennikov, Exponential groups. II: Extension of centralizers and tensor completion of csa-groups, Internat. J. Algebra Comput. 6 (1996), 687-712. CMP 97:04
  • 13. A. Yu. Ol'shanski, Periodic factor groups of hyperbolic groups, Math. USSR Sb. 72 (1992), 519-541. MR 92d:20050
  • 14. -, On residualing homomorphisms and G-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), 365-409. MR 94i:20069
  • 15. P. Papasoglu, Geometric methods in group theory, Ph.D. thesis, Columbia Univ., New York, 1993.
  • 16. M. D. Shapiro, Automatic structure and graphs of groups, Topology '90, Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 335-380. MR 93i:20044

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Additional Information

O. Kharlampovich
Affiliation: Department of Mathematics and Statistics, McGill University, Montréal, P.Q., Canada H3A 2K6

A. Myasnikov
Affiliation: Department of Mathematics, City College (CUNY), New York, New York 10031-9100

Received by editor(s): July 7, 1994
Received by editor(s) in revised form: January 18, 1996
Additional Notes: The first author was supported by grants from NSERC and FCAR
Article copyright: © Copyright 1998 American Mathematical Society

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