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On Products of Quasiconvex Subgroups in Hyperbolic Groups [article]

Ashot Minasyan
2004 arXiv   pre-print
An interesting question about 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.  ... 
arXiv:math/0404397v1 fatcat:zhtd4xifgng3fdcrjv3mfnz6ty

ON PRODUCTS OF QUASICONVEX SUBGROUPS IN HYPERBOLIC GROUPS

ASHOT MINASYAN
2004 International journal of algebra and computation  
Intersection 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.  ... 
doi:10.1142/s0218196704001712 fatcat:afrbtk4rhvchvjgglqdvl6fg3y

Quasiconvexity and relatively hyperbolic groups that split [article]

Hadi Bigdely, Daniel T. Wise
2012 arXiv   pre-print
We then provide a criterion for the relative 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.  ... 
arXiv:1211.1993v1 fatcat:qx3aa57mcbhobj6qyviwbzc7am

Separable subsets of GFERF negatively curved groups

Ashot Minasyan
2006 Journal of Algebra  
We show that 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.  ... 
doi:10.1016/j.jalgebra.2006.03.050 fatcat:dibcuqczlnbgfmptidsxrxo5pm

Separable Subsets of GFERF Negatively Curved Groups [article]

Ashot Minasyan
2006 arXiv   pre-print
We show that 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.  ... 
arXiv:math/0502101v2 fatcat:dmshxodwnnc53fqvh54dlu77lq

Detecting Relatively Quasiconvex Subgroups and Their Induced Peripheral Structure [article]

Thomas Carstensen
2022 arXiv   pre-print
This paper proves under certain conditions the existence 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.  ... 
arXiv:2203.02357v2 fatcat:lvcdpfwz3zcexmhta3xzyx2tma

Virtual amalgamation of relatively quasiconvex subgroups

Eduardo Martínez-Pedroza, Alessandro Sisto
2012 Algebraic and Geometric Topology  
The main theorem extends results for 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.  ... 
doi:10.2140/agt.2012.12.1993 fatcat:my5gxgfd2jeldkjeclluld5ndq

An example of a non-quasiconvex subgroup of a word hyperbolic group with exotic limit set [article]

Ilya Kapovich
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.  ... 
arXiv:math/9506207v1 fatcat:fxez6q27nfcxxjw4trij6k2xuu

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

Yoshifumi Matsuda, Shin-ichi Oguni, Saeko Yamagata
2012 arXiv   pre-print
We also study relations between relatively 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.  ... 
arXiv:1106.5288v3 fatcat:rdeihaqzmvhihkklxl3lpotd4e

On Quasiconvexity and Relative Hyperbolic Structures [article]

Eduardo Martinez-Pedroza
2011 arXiv   pre-print
Let G be a 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.  ... 
arXiv:0811.2384v4 fatcat:x4iqyhj5bbgw5lgeeejwwhtzay

On Residual Properties of Word Hyperbolic Groups [article]

Ashot Minasyan
2006 arXiv   pre-print
For a fixed word 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.  ... 
arXiv:math/0506606v2 fatcat:wunfj2yuc5dfjbg3lp7vgw5ngm

On residual properties of word hyperbolic groups

Ashot Minasyan
2006 Journal of group theroy  
For a fixed word 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).  ... 
doi:10.1515/jgt.2006.045 fatcat:4k2osh3oznhs7euhnufvi3rdya

Subgroup properties of fully residually free groups

Ilya Kapovich
2001 Transactions of the American Mathematical Society  
We prove that fully residually free 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.  ... 
doi:10.1090/s0002-9947-01-02840-9 fatcat:xoqcsylhkfdqribmalc5ecf5py

Research announcement: The structure of groups with a quasiconvex hierarchy

Daniel Wise
2009 Electronic Research Announcements in Mathematical Sciences  
Let G be a word-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] .  ... 
doi:10.3934/era.2009.16.44 fatcat:zhftsdi2wjf7vhwnilfasj6l5a

A non-quasiconvexity embedding theorem for hyperbolic groups [article]

Ilya Kapovich
1997 arXiv   pre-print
We show that if G is a non-elementary torsion-free word 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  ... 
arXiv:math/9704203v1 fatcat:6gxhunx5wnawbj7ldlksa44jhu
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