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Freely Indecomposable Groups Acting on Hyperbolic Spaces

Freely Indecomposable Groups Acting on Hyperbolic Spaces,10.1142/S0218196704001682,International Journal of Algebra and Computation,Ilya Kapovich,Rich

Freely Indecomposable Groups Acting on Hyperbolic Spaces   (Citations: 12)
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We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups.
Journal: International Journal of Algebra and Computation - IJAC , vol. 14, no. 2, pp. 115-171, 2004
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    • ...The finiteness of Nielsen equivalence classes of k-tuples for torsion-free locally quasiconvex-hyperbolic groups has been established by the authors [KW] generalizing a result of Delzant [D] who studied 2-generated groups...

    Ilya Kapovichet al. Nielsen equivalence in small cancellation groups

    • ...Weidmann’s Theorem 2.3 was generalized by Kapovich and Weidman in [KW04] to the setting of -hyperbolic spaces...

    Yair Glasner. A zero-one law for random subgroups of some totally disconnected group...

    • ...Moreover, Kapovich and Weidmann [38] proved that if G is a torsion-free hyperbolic group where all k-generated subgroups are quasiconvex, then G has only finitely many (up to conjugation) Nielsen-equivalence classes of (k+ 1)-tuples generating one-ended subgroups...

    Ilya Kapovichet al. Genericity, the Arzhantseva-Ol’shanskii method and the isomorphism pro...

    • ...Thus Kapovich and Weidmann [22] proved that the rank problem is solvable for torsion-free locally quasiconvex word-hyperbolic groups...
    • ...The main technical tool needed for the proof of Theorem A is machinery developed by Kapovich and Weidmann in [21, 22] that provides a far-reaching generalization of Nielsen’s methods in the general context of groups acting by isometries on Gromov-hyperbolic spaces...
    • ...For an arbitrary group G it is easy to see that rk(G) = rk(G/F(G)), where F(G) is the Frattini subgroup of G. If G is finitely generated nilpotent, then the Frattini subgroup F(G) contains the commutator [G, G]. This implies that rk(G) = rk(Gab) where Gab = G/[G, G] is the abelianization of G. The rank problem is also decidable for torsion-free word-hyperbolic locally quasiconvex groups (Kapovich–Weidmann [22]), for (sufficiently large) ...
    • ...The work of Weidmann [34] (see also Kapovich– Weidmann [22]) shows that the presence of torsion often creates a substantial difficulty for solving the rank problem...
    • ...A similar but stronger statement was obtained by Kapovich and Weidmann [22] for torsion-free one-ended locally quasiconvex hyperbolic groups...
    • ...This tool was obtained by Kapovich and Weidmann in [22]...
    • ...The following is a corollary of [22, Theorem 2.4]...
    • ...We recall here some results of Kapovich–Weidmann [22]...
    • ...Proof It is proved in [22] that c = 4c2(G, S, δ, n, K) + 3K satisfies the requirements of the proposition...
    • ...Proof It is proved in [22] that c′ = 2c2(G, S, δ, n, K) + 3K satisfies the re-...

    Ilya Kapovichet al. Kleinian groups and the rank problem

    • ...As mentioned by the referee, Proposition 3.3 is a particular case of the main technical result of Kapovich and Weidmann [8] and that it can also be derived from their [9, Theorem 2.5]...

    Juan Souto. The rank of the fundamental group of certain hyperbolic 3-manifolds fi...

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