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On Products of Quasiconvex Subgroups in Hyperbolic Groups
[article]

2004
*
arXiv
*
pre-print

An interesting question about

arXiv:math/0404397v1
fatcat:zhtd4xifgng3fdcrjv3mfnz6ty
*quasiconvexity**in*a*hyperbolic**group*concerns finding classes*of**quasiconvex*subsets that are closed under finite intersections. ...*In*this work we study the properties*of**products**of**quasiconvex**subgroups*; we show that such sets are*quasiconvex*, their finite intersections have a similar algebraic representation and, thus, are*quasiconvex*... Therefore, a left coset*of*a*quasiconvex**subgroup*and a conjugate*subgroup*to it are*quasiconvex*(*in*a*hyperbolic**group*). Lemma 2.1. ...##
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ON PRODUCTS OF QUASICONVEX SUBGROUPS IN HYPERBOLIC GROUPS

2004
*
International journal of algebra and computation
*

Intersection

doi:10.1142/s0218196704001712
fatcat:afrbtk4rhvchvjgglqdvl6fg3y
*of*two*quasiconvex**subgroups**of*a*hyperbolic**group*is, again, a*quasiconvex**subgroup*. An important property*of*a*hyperbolic**group*states that all its cyclic*subgroups*are*quasiconvex*. ... Thus, all finite unions*of*ME-*products*constitute a topology T (*of*closed sets)*on*the set*of*elements*of*a*hyperbolic**group*. ...##
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Quasiconvexity and relatively hyperbolic groups that split
[article]

2012
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arXiv
*
pre-print

We then provide a criterion for the relative

arXiv:1211.1993v1
fatcat:qx3aa57mcbhobj6qyviwbzc7am
*quasiconvexity**of*a*subgroup*H depending*on*the relative*quasiconvexity**of*the intersection*of*H with the vertex*groups**of*G. ... We explore the combination theorem for a*group*G splitting as a graph*of*relatively*hyperbolic**groups*. ... Acknowledgement: We are extremely grateful to the anonymous f.g. referee whose very helpful corrections and adjustments improved the results and exposition*of*this paper. ...##
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Separable subsets of GFERF negatively curved groups

2006
*
Journal of Algebra
*

We show that

doi:10.1016/j.jalgebra.2006.03.050
fatcat:dibcuqczlnbgfmptidsxrxo5pm
*in*any such*group*, the*product**of*finitely many*quasiconvex**subgroups*is closed*in*the profinite topology*on*G. ... A word*hyperbolic**group*G is called GFERF if every*quasiconvex**subgroup*coincides with the intersection*of*finite index*subgroups*containing it. ...*In*particular, Gitik*in*[8, Theorem 1] proved that*in*a LERF*hyperbolic**group*, a*product**of*two*quasiconvex**subgroups*,*one**of*which is malnormal, is separable. ...##
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Separable Subsets of GFERF Negatively Curved Groups
[article]

2006
*
arXiv
*
pre-print

We show that

arXiv:math/0502101v2
fatcat:dmshxodwnnc53fqvh54dlu77lq
*in*any such*group*, the*product**of*finitely many*quasiconvex**subgroups*is closed*in*the profinite topology*on*G. ... A word*hyperbolic**group*G is called GFERF if every*quasiconvex**subgroup*coincides with the intersection*of*finite index*subgroups*containing it. ...*In*particular, Gitik*in*[8, Thm. 1] proved that*in*a LERF*hyperbolic**group*, a*product**of*two*quasiconvex**subgroups*,*one**of*which is malnormal, is separable. ...##
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Detecting Relatively Quasiconvex Subgroups and Their Induced Peripheral Structure
[article]

2022
*
arXiv
*
pre-print

This paper proves under certain conditions the existence

arXiv:2203.02357v2
fatcat:lvcdpfwz3zcexmhta3xzyx2tma
*of*an algorithm, which detects relatively*quasiconvex**subgroups*H*of*relatively*hyperbolic**groups*(G,ℙ). ... Additionally, this algorithm outputs an induced peripheral structure*of*(G,ℙ)*on*H. ... A natural class*of**subgroups**of*a relatively*hyperbolic**group*(G, P) are the relatively*quasiconvex**subgroups*. ...##
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Virtual amalgamation of relatively quasiconvex subgroups

2012
*
Algebraic and Geometric Topology
*

The main theorem extends results for

doi:10.2140/agt.2012.12.1993
fatcat:my5gxgfd2jeldkjeclluld5ndq
*quasiconvex**subgroups**of*word-*hyperbolic**groups*, and results for discrete*subgroups**of*isometries*of**hyperbolic*spaces. ... For relatively*hyperbolic**groups*, we investigate conditions guaranteeing that the*subgroup*generated by two relatively*quasiconvex**subgroups*Q_1 and Q_2 is relatively*quasiconvex*and isomorphic to Q_1 ... Acknowledgments We would like to thank Wen-yuan Yang for corrections*on*an earlier version*of*the paper, and for pointing out Corollary 6. ...##
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An example of a non-quasiconvex subgroup of a word hyperbolic group with exotic limit set
[article]

1995
*
arXiv
*
pre-print

*In*particular, this gives a counterexample to the conjecture

*of*G.Swarup that a finitely presented

*one*-ended

*subgroup*

*of*a word

*hyperbolic*

*group*is

*quasiconvex*if and only if it has finite index

*in*its ... We construct an example

*of*a torsion free freely indecomposable finitely presented non-

*quasiconvex*

*subgroup*H

*of*a word

*hyperbolic*

*group*G such that the limit set

*of*H is not the limit set

*of*a

*quasiconvex*...

*In*this note we construct an example

*of*a non-

*quasiconvex*finitely presented

*one*-ended

*subgroup*H

*of*a word

*hyperbolic*

*group*G such that the limit set

*of*H is exotic. ...

##
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The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups
[article]

2012
*
arXiv
*
pre-print

We also study relations between relatively

arXiv:1106.5288v3
fatcat:rdeihaqzmvhihkklxl3lpotd4e
*hyperbolic*structures*on*a*group*and relative*quasiconvexity*for*subgroups**of*the*group*. ... We discuss the notion*of*the universal relatively*hyperbolic*structure*on*a*group*which is used*in*order to characterize relatively*hyperbolic*structures*on*the*group*. ... We also study relations between relatively*hyperbolic*structures*on*a*group*and relative*quasiconvexity*for*subgroups**of*the*group*. ...##
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On Quasiconvexity and Relative Hyperbolic Structures
[article]

2011
*
arXiv
*
pre-print

Let G be a

arXiv:0811.2384v4
fatcat:x4iqyhj5bbgw5lgeeejwwhtzay
*group*which is*hyperbolic*relative to a collection*of**subgroups*A, and it is also*hyperbolic*relative to a collection*of**subgroups*B. Suppose that the collection A contains B. ... We characterize, for*subgroups**of*G, when quasicovexity relative to A implies*quasiconvexity*relative to B. We also show that*quasiconvexity*relative to B implies*quasiconvexity*relative to A. ... notion*of**quasiconvexity**in*word*hyperbolic**groups*. ...##
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On Residual Properties of Word Hyperbolic Groups
[article]

2006
*
arXiv
*
pre-print

For a fixed word

arXiv:math/0506606v2
fatcat:wunfj2yuc5dfjbg3lp7vgw5ngm
*hyperbolic**group*we compare different residual properties related to*quasiconvex**subgroups*. ... Thus, a conjugate*of*a*quasiconvex**subgroup**in*a*hyperbolic**group*is*quasiconvex*itself. ... Free*products**of*GFERF*groups**In*the previous section we considered*hyperbolic**groups*which engulf every proper*quasiconvex**subgroup*. Let us name them QE-*groups*, for brevity. ...##
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On residual properties of word hyperbolic groups

2006
*
Journal of group theroy
*

For a fixed word

doi:10.1515/jgt.2006.045
fatcat:4k2osh3oznhs7euhnufvi3rdya
*hyperbolic**group*we compare different residual properties related to*quasiconvex**subgroups*. 2000 Mathematics Subject Classification. Primary 20F67, Secondary 20E26. ... Free*products**of*GFERF*groups**In*the previous section we considered*hyperbolic**groups*which engulf every proper*quasiconvex**subgroup*. Let us name them QE-*groups*, for brevity. ... A*hyperbolic**group*is called GFERF if each*quasiconvex**subgroup*is closed*in*PT (G). ...##
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Subgroup properties of fully residually free groups

2001
*
Transactions of the American Mathematical Society
*

We prove that fully residually free

doi:10.1090/s0002-9947-01-02840-9
fatcat:xoqcsylhkfdqribmalc5ecf5py
*groups*have the Howson property, that is the intersection*of*any two finitely generated*subgroups**in*such a*group*is again finitely generated. ... We also establish some commensurability properties for finitely generated fully residually free*groups*which are similar to those*of*free*groups*. ... Infinite cyclic*subgroups**of*word-*hyperbolic**groups*are always*quasiconvex*[1] , so L is*quasiconvex**in*A. ...##
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Research announcement: The structure of groups with a quasiconvex hierarchy

2009
*
Electronic Research Announcements in Mathematical Sciences
*

Let G be a word-

doi:10.3934/era.2009.16.44
fatcat:zhftsdi2wjf7vhwnilfasj6l5a
*hyperbolic**group*with a*quasiconvex*hierarchy. We show that G has a finite index*subgroup*G that embeds as a*quasiconvex**subgroup**of*a right-angled Artin*group*. ... It follows that every*quasiconvex**subgroup**of*G is a virtual retract, and is hence separable. The results are applied to certain 3-manifold and*one*-relator*groups*. ... The hierarchy terminates at a virtually free*group**of*the form Z n * F . For*one*-relator*groups*with torsion, we show that the*subgroups*M, M are*quasiconvex*at each level*of*the hierarchy*in*[47] . ...##
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A non-quasiconvexity embedding theorem for hyperbolic groups
[article]

1997
*
arXiv
*
pre-print

We show that if G is a non-elementary torsion-free word

arXiv:math/9704203v1
fatcat:6gxhunx5wnawbj7ldlksa44jhu
*hyperbolic**group*then there exists another word*hyperbolic**group*G^*, such that G is a*subgroup**of*G^* but G is not*quasiconvex**in*G^*. ...*Quasiconvex**subgroups*are*of*particularly great importance when*one*studies amalgamated free*products*and HNN-extensions*of**hyperbolic**groups*[Gr] , [BF1] , [KM] , [BGSS] , [Gi] , [Pa] . ... Roughly speaking, a finitely generated*subgroup*H*of*a word*hyperbolic**group*G is*quasiconvex**in*G if for any word metric d G*on*G and any word metric d H*on*H the inclusion map i : (H, d H ) −→ (G, d ...
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