A note on groups with separable finitely generated subgroups

Burns, R. G.; Karrass, A.; Solitar, D.

doi:10.1017/s0004972700026393

Cited by 58 publications

(74 citation statements)

References 7 publications

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“…One is derived from the recent work of Rubinstein and Wang, [6], and we consider it in Theorem 1. The other was the first known example of a 3-manifold group which failed to be subgroup separable and was introduced in [1] and further studied in [4] and [5]. Our proof that it fails to satisfy the engulfing property is more elementary than the original proof that it fails subgroup separability, and we hope that it sheds some light on this fact.…”

Section: Introductionmentioning

confidence: 87%

The engulfing property for 3–manifolds

Niblo¹,

Wise²

1998

Geometry &Amp; Topology Monographs

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Section: Introductionmentioning

confidence: 87%

The engulfing property for 3–manifolds

Niblo¹,

Wise²

1998

Geometry &Amp; Topology Monographs

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“…This raises the question of which 3-manifolds have LERF fundamental group. There are examples of certain graph 3-manifolds M for which π 1 (M ) is not LERF [10]. But it is conjectured that the fundamental group of every closed hyperbolic 3-manifold is LERF.…”

Section: Subgroup Separability Special Cube Complexes and Virtual Fimentioning

confidence: 99%

Finite Covering Spaces of 3-manifolds

Lackenby

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. Following Perelman's solution to the Geometrisation Conjecture, a 'generic' closed 3-manifold is known to admit a hyperbolic structure. However, our understanding of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyperbolic 3-manifold has a finite cover that contains a closed embedded orientable π1-injective surface with positive genus.I will give a survey on the progress towards this conjecture and its variants. Along the way, I will address other interesting questions, including: What are the main types of finite covering space of a hyperbolic 3-manifold? How many are there, as a function of the covering degree? What geometric, topological and algebraic properties do they have? I will show how an understanding of various geometric and topological invariants (such as the first eigenvalue of the Laplacian, the rank of mod p homology and the Heegaard genus) can be used to deduce the existence of π1-injective surfaces, and more.

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“…The first example is ht; a; b j tat 1 D ab; tbt 1 D bi due to Burns, Karrass and Solitar in [24] and is also famous for being the first 3-manifold group known not to be LERF. Also Corollary 3 of [55] gives further examples.…”

Section: Theorem 31 If G Is Lerf and Of Deficiency 1 Then Either G mentioning

confidence: 99%

Largeness of LERF and 1-relator groups

Button¹

2010

Groups Geom. Dyn.

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Abstract. We consider largeness of groups given by a presentation of deficiency 1, where the group is respectively free-by-cyclic, LERF or 1-relator. We give the first examples of (finitely generated free)-by-Z word hyperbolic groups which are large, show that a LERF deficiency 1 group with first Betti number at least two is large or Z Z and show that 2-generator 1-relator groups where the relator has height 1 obey the dichotomy that either the group is large or all its finite images are metacyclic. Mathematics Subject Classification (2010). 20F05.

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A note on groups with separable finitely generated subgroups

Cited by 58 publications

References 7 publications

The engulfing property for 3–manifolds

The engulfing property for 3–manifolds

Finite Covering Spaces of 3-manifolds

Largeness of LERF and 1-relator groups

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