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Keywords
(4)
Conjugacy Class
Hyperbolic Group
Hyperbolic Space
Word Hyperbolic Group
Related Publications
(4)
Nielsen methods and groups acting on hyperbolic spaces
Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations
Doubles of groups and hyperbolic LERF 3manifolds
Kleinian groups and the rank problem
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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
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Freely Indecomposable Groups Acting on Hyperbolic Spaces
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Citations: 12
)
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Ilya Kapovich
,
Richard Weidmann
We obtain a number of finiteness results for groups acting on Gromovhyperbolic spaces. In particular we show that a torsionfree locally quasiconvex
hyperbolic group
has only finitely many conjugacy classes of ngenerated oneended subgroups.
Journal:
International Journal of Algebra and Computation  IJAC
, vol. 14, no. 2, pp. 115171, 2004
DOI:
10.1142/S0218196704001682
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Citation Context
(10)
...The finiteness of Nielsen equivalence classes of ktuples for torsionfree locally quasiconvexhyperbolic groups has been established by the authors [
KW
] generalizing a result of Delzant [D] who studied 2generated groups...
Ilya Kapovich
,
et 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 zeroone law for random subgroups of some totally disconnected group...
...Moreover, Kapovich and Weidmann [
38
] proved that if G is a torsionfree hyperbolic group where all kgenerated subgroups are quasiconvex, then G has only finitely many (up to conjugation) Nielsenequivalence classes of (k+ 1)tuples generating oneended subgroups...
Ilya Kapovich
,
et al.
Genericity, the ArzhantsevaOl’shanskii method and the isomorphism pro...
...Thus Kapovich and Weidmann [
22
] proved that the rank problem is solvable for torsionfree locally quasiconvex wordhyperbolic 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 farreaching generalization of Nielsen’s methods in the general context of groups acting by isometries on Gromovhyperbolic 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 torsionfree wordhyperbolic 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 torsionfree oneended 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 Kapovich
,
et 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 3manifolds fi...
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(26)
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(
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(
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Journal:
Journal of Pure and Applied Algebra  J PURE APPL ALG
, vol. 89, no. 12, pp. 347, 1993
Unsolvable Problems About Small Cancellation and Word Hyperbolic Groups
(
Citations: 18
)
G. Baumslag
,
C. F. Miller
,
H. Short
Journal:
Bulletin of The London Mathematical Society  BULL LOND MATH SOC
, vol. 26, no. 1, pp. 97101, 1994
Boundaries of hyperbolic groups
(
Citations: 25
)
Ilya Kapovich
,
Nadia Benakli
Published in 2002.
A combination theorem for negatively curved groups
(
Citations: 140
)
M. Bestvina
,
M. Feighn
Journal:
Journal of Differential Geometry  J DIFFEREN GEOM
, vol. 35, no. 1992, pp. 85101, 1992
Sort by:
Citations
(12)
Nielsen equivalence in small cancellation groups
Ilya Kapovich
,
Richard Weidmann
Published in 2010.
A zeroone law for random subgroups of some totally disconnected groups
(
Citations: 1
)
Yair Glasner
Journal:
Transformation Groups  TRANSFORM GROUPS
, vol. 14, no. 4, pp. 787800, 2009
Conjugacy of 2–spherical subgroups of Coxeter groups and parallel walls
(
Citations: 7
)
PierreEmmanuel Caprace
Journal:
Algebraic and Geometric Topology  ALGEBR GEOM TOPOL
, vol. 6, pp. 19872029, 2006
Genericity, the ArzhantsevaOl’shanskii method and the isomorphism problem for onerelator groups
(
Citations: 20
)
Ilya Kapovich
,
Paul Schupp
Journal:
Mathematische Annalen  MATH ANN
, vol. 331, no. 1, pp. 119, 2005
Kleinian groups and the rank problem
(
Citations: 7
)
Ilya Kapovich
,
Richard Weidmann
Journal:
Geometry & Topology  GEOM TOPOL
, vol. 9, pp. 375402, 2005