scholarly journals Abelian subgroup separability of some one-relator groups

2005 ◽  
Vol 2005 (14) ◽  
pp. 2287-2298 ◽  
Author(s):  
D. Tieudjo
Keyword(s):  
Abelian Subgroup ◽  
Cyclic Subgroup ◽  
Finitely Generated ◽  
One Relator Groups ◽  
Subgroup Separability

We prove that any group in the class of one-relator groups given by the presentation〈a,b;[am,bn]=1〉, wheremandnare integers greater than 1, is cyclic subgroup separable (orπc). We establish some suitable properties of these groups which enable us to prove that every finitely generated abelian subgroup of any of such groups is finitely separable.

Abelian subgroup separability of certain HNN extensions

2018 ◽  
Vol 28 (03) ◽  
pp. 543-552
Author(s):  
Wei Zhou ◽  
Goansu Kim
Keyword(s):  
Abelian Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Hnn Extensions ◽  
Subgroup Separability

In this paper, we prove that certain HNN extensions of finitely generated abelian subgroup separable groups are finitely generated abelian subgroup separable. Using this, we show that certain HNN extensions of finitely generated nilpotent groups are finitely generated abelian subgroup separable.

scholarly journals On the cyclic subgroup separability of the free product of two groups with commuting subgroups

2014 ◽  
Vol 24 (05) ◽  
pp. 741-756 ◽  
Author(s):  
E. V. Sokolov
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups A and B with commuting subgroups H ≤ A and K ≤ B, and let 𝒞 be the class of all finite groups or the class of all finite p-groups. We derive the description of all 𝒞-separable cyclic subgroups of G provided this group is residually a 𝒞-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p′-isolated in the free factors, then all p′-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p′-isolated and cyclic.

Subgroup Separability of Generalized Free Products of Free-By-Finite Groups

1993 ◽  
Vol 36 (4) ◽  
pp. 385-389 ◽  
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang
Keyword(s):  
Finite Groups ◽  
Word Problems ◽  
Cyclic Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Subgroup Separability ◽  
Solvable Word ◽  
Generalized Free Products

AbstractWe prove that generalized free products of finitely generated free-byfinite groups amalgamating a cyclic subgroup are subgroup separable. From this it follows that if where t ≥ 1 and u, v are words on {a1,...,am} and {b1,...,bn} respectively then G is subgroup separable thus generalizing a result in [9] that such groups have solvable word problems.

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

QUASI-ISOMETRIES AND AMALGAMATIONS OF TAME COMBABLE GROUPS

1995 ◽  
Vol 05 (02) ◽  
pp. 199-204 ◽  
Author(s):  
STEPHEN G. BRICK
Keyword(s):  
Analogous Result ◽  
Finitely Generated ◽  
Hnn Extensions ◽  
One Relator Groups

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A *C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

scholarly journals A note on groups with separable finitely generated subgroups

1987 ◽  
Vol 36 (1) ◽  
pp. 153-160 ◽  
Author(s):  
R. G. Burns ◽  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Cyclic Extension ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Positive Direction ◽  
One Relator Groups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

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