Pro-p completions of groups of cohomological dimension 2

2016 ◽  
Vol 26 (03) ◽  
pp. 551-564
Author(s):  
Dessislava H. Kochloukova
Keyword(s):  
Free Group ◽  
Cohomological Dimension ◽  
Finitely Generated ◽  
Limit Group ◽  
Finitely Presented Group ◽  
Dimension Formula ◽  
Dimension 2 ◽  
Finitely Presented

We study when an abstract finitely presented group [Formula: see text] of cohomological dimension [Formula: see text] has pro-[Formula: see text] completion [Formula: see text] of cohomological dimension [Formula: see text]. Furthermore, we prove that for a tree hyperbolic limit group [Formula: see text] we have [Formula: see text] and show an example of a hyperbolic limit group [Formula: see text] that is not free and [Formula: see text] is free pro-[Formula: see text]. For a finitely generated residually free group [Formula: see text] that is not a limit group, we show that [Formula: see text] is not free pro-[Formula: see text].

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

Free Group Automorphisms and Knotted Tori in S4

1997 ◽  
Vol 06 (01) ◽  
pp. 95-103 ◽  
Author(s):  
Daniel S. Silver
Keyword(s):  
Free Group ◽  
Commutator Subgroup ◽  
Sufficient Conditions ◽  
Simple Construction ◽  
Finitely Generated ◽  
Seifert Manifolds ◽  
Group Automorphisms ◽  
Free Group Automorphisms ◽  
Finitely Presented

Let ϕ be an automorphism of a finitely generated free group F, w and element of F, and H the subgroup of F generated by the orbit {ϕn (w), |n ∈ Z}. We describe sufficient conditions ensuring that H is non-finitely generated. Using this we give a simple construction of tori T embedded in S4 in such a way that the commutator subgroup of π1(S4 - T) is finitely generated but not finitely presented. Such tori have no minimal Seifert manifolds. An example of an embedded torus with the latter property was given recently by T. Maeda using different methods.

A group with zero homology

1968 ◽  
Vol 64 (3) ◽  
pp. 599-602 ◽  
Author(s):  
D. B. A Epstein
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Dimensional Manifold ◽  
Finitely Generated ◽  
3 Dimensional ◽  
Identity Group ◽  
Dimension 2

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

Whitehead Graphs and the Σ1-Invariants of Infinite Groups

1998 ◽  
Vol 08 (01) ◽  
pp. 23-34 ◽  
Author(s):  
Susan Garner Garille ◽  
John Meier
Keyword(s):  
Local Structure ◽  
Universal Cover ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Group ◽  
Finite Presentation ◽  
Finitely Presented Group ◽  
Finitely Presented

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ1(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ1(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ1(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ1(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.

Computing with knot quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850074
Author(s):  
Graham Ellis ◽  
Cédric Fragnaud
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Knot Invariant ◽  
Number Formula ◽  
Finitely Presented

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

scholarly journals LIMIT GROUPS ARE CONJUGACY SEPARABLE

2007 ◽  
Vol 17 (04) ◽  
pp. 851-857 ◽  
Author(s):  
S. C. CHAGAS ◽  
P. A. ZALESSKII
Keyword(s):  
Free Group ◽  
Finitely Generated ◽  
Limit Group ◽  
Limit Groups ◽  
Conjugacy Separable

A limit group is a finitely-generated subgroup of a fully residually free group. We prove in this paper the result announced in the title.

On groups and 3-manifolds with weak dimension ≤ 1

1974 ◽  
Vol 75 (1) ◽  
pp. 33-35
Author(s):  
David E. Galewski
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Connected Sum ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Weak Dimension ◽  
Finitely Presented

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

scholarly journals INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 335-344
Author(s):  
MUSTAFA GÖKHAN BENLI
Keyword(s):  
Normal Subgroup ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Free Semigroups ◽  
Finitely Presented ◽  
Indicable Group

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

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