Deficiency and abelianized deficiency of some virtually free groups

AbstractLet Qm be the HNN extension of Z/m × Z/m where the stable letter conjugates the first factor to the second. We explore small presentations of the groups Γm,n = Qm* Qn. We show that for certain choices of (m,n), for example (2,3), the group Γm,n has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if Γm,n has a relation gap.

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VIRTUAL PROPERTIES OF CYCLICALLY PINCHED ONE-RELATOR GROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196709005032 ◽

2009 ◽

Vol 19 (02) ◽

pp. 213-227 ◽

Cited By ~ 2

Author(s):

GILBERT BAUMSLAG ◽

BENJAMIN FINE ◽

CHARLES F. MILLER ◽

DOUGLAS TROEGER

Keyword(s):

Free Group ◽

Cyclic Subgroup ◽

Free Groups ◽

Hnn Extensions ◽

Hnn Extension ◽

One Relator Groups ◽

Special Case ◽

Torsion Free ◽

Amalgamated Product

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general, the amalgamated product of free groups with cyclic amalgamations is free-by-(torsion-free nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-by-abelian. Analagous results are obtained for certain one-relator HNN-extensions.

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ASCENDING HNN EXTENSIONS OF FINITELY GENERATED FREE GROUPS ARE HOPFIAN

Bulletin of the London Mathematical Society ◽

10.1017/s0024609301007986 ◽

2001 ◽

Vol 33 (3) ◽

pp. 292-298 ◽

Cited By ~ 10

Author(s):

ROSS GEOGHEGAN ◽

MICHAEL L. MIHALIK ◽

MARK SAPIR ◽

DANIEL T. WISE

Keyword(s):

Free Group ◽

Free Groups ◽

Finitely Generated ◽

Important Ingredient ◽

Hnn Extensions ◽

Hnn Extension

It is shown that every ascending HNN extension of a finitely generated free group is Hopfian. An important ingredient in the proof is that under certain hypotheses on the group H, if G is an ascending HNN extension of H, then cd(G) = cd(H) + 1.

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On 3-strand singular pure braid group

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520420018 ◽

2020 ◽

Vol 29 (10) ◽

pp. 2042001

Author(s):

Valeriy G. Bardakov ◽

Tatyana A. Kozlovskaya

Keyword(s):

Direct Product ◽

Braid Group ◽

Product Formula ◽

Direct Factor ◽

Defining Relations ◽

Pure Braid Group ◽

Hnn Extension ◽

Base Group

In this paper, we study the singular pure braid group [Formula: see text] for [Formula: see text]. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that [Formula: see text] is a semi-direct product [Formula: see text], where [Formula: see text] is an HNN-extension with base group [Formula: see text] and cyclic associated subgroups. We prove that the center [Formula: see text] of [Formula: see text] is a direct factor in [Formula: see text].

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Discrete Groups of Motions

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-036-8 ◽

1960 ◽

Vol 12 ◽

pp. 415-426 ◽

Cited By ~ 48

Author(s):

Leon Greenberg

Keyword(s):

Hyperbolic Plane ◽

Free Groups ◽

Discrete Groups ◽

Finitely Generated ◽

Defining Relations ◽

Rigid Motions ◽

Image Position

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:Generators: .Defining relations: where km = ambmam-1bm-1. We shall denote this group by F(p; n1, … , nd; r).In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.

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Sulphoamidic Phtalocyaninic Pigments and Dyes(II)

Revista de Chimie ◽

10.37358/rc.08.5.1820 ◽

2008 ◽

Vol 59 (5) ◽

Author(s):

Elena Stingaciu ◽

Corneliu Minca ◽

Ion Sebe

Keyword(s):

Ir Spectra ◽

Aromatic Amines ◽

Free Groups ◽

Cyanuric Chloride ◽

Layer Chromatography ◽

Phenylene Diamine

This work concerns the synthesis of pigments and phtalocyanine dyes obtained through the sulphonation of copper phtalocyanine and amidation with some aliphatic and aromatic amines (lauryl-amine, i-propyl-amine, hexadecyl-amine, stearyl-amine and acetyl-p-phenylene-diamine) with good properties for the electrotechnic utilisation and for toner materials. The pigments with amino free groups are transformed by condensation with cyanuric chloride in phtalocyanine pigments with different tinctorial properties. The dyes were analyzed through the layer chromatography and were characterized on the IR spectra bases and tinctorial tests.

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Generating the inverse limit of free groups

Journal of Algebra ◽

10.1016/j.jalgebra.2021.02.017 ◽

2021 ◽

Vol 578 ◽

pp. 371-401

Author(s):

Gregory R. Conner ◽

Wolfgang Herfort ◽

Curtis A. Kent ◽

Petar Pavešić

Keyword(s):

Inverse Limit ◽

Free Groups

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Minimal partition-free groups

Ricerche di Matematica ◽

10.1007/s11587-021-00579-z ◽

2021 ◽

Author(s):

Afsane Bahri ◽

Zeinab Akhlaghi ◽

Behrooz Khosravi

Keyword(s):

Free Groups

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Freely indecomposable almost free groups with free abelianization

Journal of Group Theory ◽

10.1515/jgth-2019-0102 ◽

2020 ◽

Vol 23 (3) ◽

pp. 531-543

Author(s):

Samuel M. Corson

Keyword(s):

Free Groups ◽

Uncountable Cardinals

AbstractFor certain uncountable cardinals κ, we produce a group of cardinality κ which is freely indecomposable, strongly κ-free, and whose abelianization is free abelian of rank κ. The construction takes place in Gödel’s constructible universe L. This strengthens an earlier result of Eklof and Mekler.

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