Hyperbolic groups and free constructions

Kharlampovich, O.; Myasnikov, A.

doi:10.1090/S0002-9947-98-01773-5

Hyperbolic groups and free constructions
HTML articles powered by AMS MathViewer

by O. Kharlampovich and A. Myasnikov

Trans. Amer. Math. Soc. 350 (1998), 571-613

DOI: https://doi.org/10.1090/S0002-9947-98-01773-5

PDF | Request permission

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 $G^{\mathbf {Q}}$.

References

Gilbert Baumslag, On free ${\cal D}$-groups, Comm. Pure Appl. Math. 18 (1965), 25–30. MR 177028, DOI 10.1002/cpa.3160180105
Gilbert Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217–303. MR 122859, DOI 10.1007/BF02546390
G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991), no. 3, 229–316. MR 1147304, DOI 10.1016/0022-4049(91)90139-S
G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams—a survey, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 179–194. MR 1230634, DOI 10.1007/978-1-4613-9730-4_{8}
M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226, DOI 10.4310/jdg/1214447806
S. M. Gersten and H. B. Short, Rational subgroups of biautomatic groups, Ann. of Math. (2) 134 (1991), no. 1, 125–158. MR 1114609, DOI 10.2307/2944334
R. Gitik, On combination theorems for negatively curved groups, Internat. J. Algebra Comput. 6 (1996), 751–760.
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
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).
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
K. V. Mikhajlovskiĭ and A. Yu. Ol′shanskiĭ, Some constructions relating to hyperbolic groups, 1994, Proc. Int. Conf. on Cohomological and Geometric Methods in Group Theory (to appear).
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.
A. Yu. Ol′shanskiĭ, Periodic quotient groups of hyperbolic groups, Mat. Sb. 182 (1991), no. 4, 543–567 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 519–541. MR 1119008
A. Yu. Ol′shanskiĭ, On residualing homomorphisms and $G$-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365–409. MR 1250244, DOI 10.1142/S0218196793000251
P. Papasoglu, Geometric methods in group theory, Ph.D. thesis, Columbia Univ., New York, 1993.
Michael Shapiro, Automatic structure and graphs of groups, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 355–380. MR 1184421