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Hyperbolic groups and free constructions. (English) Zbl 0902.20018
The authors make the following definitions: A subgroup $$U$$ of a group $$G$$ is said to be conjugate separated if the set $$\{u\in U\mid u^x\in U\}$$ is finite for all $$x\in G\setminus U$$. Suppose now that $$U$$ and $$V$$ are subgroups of $$G$$, let $$\psi\colon U\to V$$ be an isomorphism, that either $$U$$ or $$V$$ is conjugate separated and that the set $$\{U\cap g^{-1}Vg\}$$ is finite for all $$g\in G$$. Then, the HNN-extension $$\langle G,t\mid t^{-1}ut=u^\psi,\;u\in U\rangle$$ is said to be separated.
The authors then prove the following Theorem 1. If $$G$$ is a hyperbolic group and $$H=\langle G,t\mid U^t=V\rangle$$ is a separated HNN-extension such that the subgroups $$U$$ and $$V$$ are quasiisometrically embedded in $$G$$, then $$H$$ is hyperbolic. – Theorem 2. Let $$G_1$$ and $$G_2$$ be hyperbolic groups, with $$U\leq G_1$$ and $$V\leq G_2$$ quasiisometrically embedded, and $$U$$ conjugate separated in $$G_1$$. Then the group $$G_1*_{U=V}G_2$$ is hyperbolic.
They obtain the following corollaries: Corollary 1. If $$G$$ is a hyperbolic group with $$A$$ and $$B$$ isomorphic virtually cyclic subgroups, then the HNN-extension $$H=\langle G,t\mid A^t=B\rangle$$ is hyperbolic if and only if it is separated. – Corollary 2. Let $$G_1$$ and $$G_2$$ be hyperbolic groups, with $$A\leq G_1$$, $$B\leq G_2$$ virtually cyclic. Then the group $$G_1*_{A=B}G_2$$ is hyperbolic if and only if either $$A$$ is conjugate separated in $$G_1$$ or $$B$$ is conjugate separated in $$G_2$$. Corollary 2 has been proved by M. Bestvina and M. Feighn [in J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029)]. The authors note that Corollary 1 contradicts an assertion in this same paper.
The authors study then quasiconvexity, and they prove the following Theorem 3. Let $$H=\langle G,t\mid U^t=V\rangle$$ be hyperbolic with $$U$$ quasiconvex in $$H$$. Then, $$G$$ is quasiconvex in $$H$$ and hence hyperbolic. – Theorem 4. Let $$H$$ be a separated HNN-extension, $$H=\langle G,t\mid U^t=V\rangle$$, with $$G$$ hyperbolic and $$U$$ and $$V$$ quasiconvex in $$G$$. Then $$G$$ is quasiconvex in $$H$$.
Finally, the authors describe the $$Q$$-completion $$G^Q$$ of a torsion-free hyperbolic group $$G$$ (where $$Q$$ is the field of rationals) as a union of an effective chain of hyperbolic subgroups, and they give a solution for the word problem and the conjugacy problem in $$G^Q$$.

MSC:
 20F65 Geometric group theory 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 57M07 Topological methods in group theory
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References:
 [1] Gilbert Baumslag, On free \cal\?-groups, Comm. Pure Appl. Math. 18 (1965), 25 – 30. · Zbl 0136.01204 · doi:10.1002/cpa.3160180105 · doi.org [2] Gilbert Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217 – 303. · Zbl 0178.34901 · doi:10.1007/BF02546390 · doi.org [3] G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991), no. 3, 229 – 316. · Zbl 0749.20006 · doi:10.1016/0022-4049(91)90139-S · doi.org [4] 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. · Zbl 0749.20017 · doi:10.1007/978-1-4613-9730-4_8 · doi.org [5] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85 – 101. · Zbl 0724.57029 [6] S. M. Gersten and H. B. Short, Rational subgroups of biautomatic groups, Ann. of Math. (2) 134 (1991), no. 1, 125 – 158. · Zbl 0744.20035 · doi:10.2307/2944334 · doi.org [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, New York, 1987, pp. 75 – 263. · Zbl 0634.20015 · doi:10.1007/978-1-4613-9586-7_3 · doi.org [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] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. · Zbl 0138.25604 [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 · Zbl 0866.20014 [13] A. Yu. Ol$$^{\prime}$$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. [14] A. Yu. Ol$$^{\prime}$$shanskiĭ, On residualing homomorphisms and \?-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365 – 409. · Zbl 0830.20053 · doi:10.1142/S0218196793000251 · doi.org [15] P. Papasoglu, Geometric methods in group theory, Ph.D. thesis, Columbia Univ., New York, 1993. [16] 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. · Zbl 0798.20020
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