Limit groups are subgroup conjugacy separable

Chagas, S. C.; Zalesskiĭ, Pavel

doi:10.1016/j.jalgebra.2016.04.026

Cited by 8 publications

(4 citation statements)

References 21 publications

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“…Bux in [5] proved that surface groups are conjugacy subgroup separable. In [10] the authors of the present paper extended this result to limit groups.…”

Section: Introductionmentioning

confidence: 60%

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

Chagas¹,

Zalesskiĭ²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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“…Bux in [5] proved that surface groups are conjugacy subgroup separable. In [10] the authors of the present paper extended this result to limit groups.…”

Section: Introductionmentioning

confidence: 60%

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

Chagas¹,

Zalesskiĭ²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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“…We used Theorem A to extend subgroup conjugacy separability to a class of groups including surface groups (see [4]). Meanwhile, Chagas and Zalesskii [6] had generalized our result about surface groups in a different direction showing that limit groups are subgroup conjugacy separable. They use different methods.…”

Section: Introductionmentioning

confidence: 90%

From local to global conjugacy of subgroups of relatively hyperbolic groups

Bogopolski

Bux

2017

Int. J. Algebra Comput.

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Abstract. Suppose that a finitely generated group G is hyperbolic relative to a collection of subgroups P = {P1, . . . , Pm}. Let H1, H2 be subgroups of G such that H1 is relatively quasiconvex with respect to P and H2 is not parabolic. Suppose that H2 is elementwise conjugate into H1. Then there exists a finite index subgroup of H2 which is conjugate into H1. The minimal length of the conjugator can be estimated.In the case where G is a limit group, it is sufficient to assume only that H1 is a finitely generated and H2 is an arbitrary subgroup of G.

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“…Remark. (1) An alternative proof of a variant of Corollary C is given by Chagas and Zalesskii in [9,Theorem 1.2].…”

Section: Introductionmentioning

confidence: 99%

“…(4) From ( 2) and (3), and using Proposition 8.1, we get that the groups F n × F m with n, m 2 are not cyclic-SICS and not SCS. (An alternative proof that F 2 × F 2 is not SCS can be found in [9].) The latter stands in contrast to the fact that these groups are conjugacy separable.…”

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confidence: 99%

On subgroup conjugacy separability of hyperbolic QVH-groups

Bogopolski,

Bux

2016

Preprint

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A group G is called subgroup conjugacy separable (abbreviated as SCS) if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. An into-conjugacy version of SCS is abbreviated by SICS. We prove that if G is a hyperbolic group, H1 is a quasiconvex subgroup of G, and H2 is a subgroup of G which is elementwise conjugate into H1, then there exists a finite index subgroup of H2 which is conjugate into H1. As corollary, we deduce that fundamental groups of closed hyperbolic 3-manifolds and torsion-free small cancellation groups with finite C ′ (1/6) or C ′ (1/4) − T (4) presentations are hereditarily quasiconvex-SCS and hereditarily quasiconvex-SICS, and that surface groups are SCS and SICS. We also show that the word "quasiconvex" cannot be deleted for at least small cancellation groups.This property logically continues the following series of well known properties of groups: residual finiteness, conjugacy separability (CS), and subgroup separability (LERF). Note that SCS-groups are residually finite, but there are residually finite, and even conjugacy separable groups, which are not SCS-groups. The SCS-property is relatively new and not much is known about, which groups enjoy this property.In [16], Grunewald and Segal proved that all virtually polycyclic groups are SCS (see also Theorem 7 in Chapter 4 of [39]). In the preprint [6], Bogopolski and Grunewald proved that free groups and some virtually free groups are SCS. In the preprint [2], Bogopolski and Bux proved that the fundamental groups of closed orientable surfaces are SCS. Chagas and Zalesskii [9] generalized this to limit groups. To formulate our results, we need the following definitions:

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Limit groups are subgroup conjugacy separable

Cited by 8 publications

References 21 publications

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

From local to global conjugacy of subgroups of relatively hyperbolic groups

On subgroup conjugacy separability of hyperbolic QVH-groups

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