Finitely presented subgroups of a product of two free groups

2001 ◽  
Vol 52 (1) ◽  
pp. 127-131
Author(s):  
H. Short
Keyword(s):  
Free Groups ◽  
Finitely Presented

scholarly journals Inversion is Possible in Groups with no Periodic Automorphisms

2015 ◽  
Vol 59 (1) ◽  
pp. 11-16
Author(s):  
Martin R. Bridson ◽  
Hamish Short
Keyword(s):  
Free Groups ◽  
Trivial Element ◽  
Finitely Presented ◽  
Torsion Free

AbstractThere exist infinite finitely presented torsion-free groups G such that Aut(G) and Out(G) are torsion free but G has an automorphism sending some non-trivial element to its inverse.

scholarly journals A periodicity theorem for acylindrically hyperbolic groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Bogopolski
Keyword(s):  
Free Groups ◽  
Hyperbolic Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Periodicity Theorem ◽  
Finitely Presented

AbstractWe generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

On the Bieri–Neumann–Strebel–Renz invariants of residually free groups

2020 ◽  
Vol 63 (3) ◽  
pp. 807-829
Author(s):  
Dessislava H. Kochloukova ◽  
Francismar Ferreira Lima
Keyword(s):  
Free Groups ◽  
Finite Union ◽  
Higher Dimensional ◽  
Homological Invariants ◽  
Finitely Presented

AbstractWe calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

scholarly journals ACTIONS, LENGTH FUNCTIONS, AND NON-ARCHIMEDEAN WORDS

2013 ◽  
Vol 23 (02) ◽  
pp. 325-455 ◽  
Author(s):  
OLGA KHARLAMPOVICH ◽  
ALEXEI MYASNIKOV ◽  
DENIS SERBIN
Keyword(s):  
Free Groups ◽  
Length Functions ◽  
Methods And Techniques ◽  
Recent Developments ◽  
Finitely Presented

In this paper we survey recent developments in the theory of groups acting on Λ-trees. We are trying to unify all significant methods and techniques, both classical and recently developed, in an attempt to present various faces of the theory and to show how these methods can be used to solve major problems about finitely presented Λ-free groups. Besides surveying results known up to date we draw many new corollaries concerning structural and algorithmic properties of such groups.

Finitely presented extensions by free groups

2007 ◽  
Vol 10 (5) ◽  
Author(s):  
Gilbert Baumslag ◽  
Charles F Miller
Keyword(s):  
Free Groups ◽  
Finitely Presented

scholarly journals Reidemeister moves and groups

2015 ◽  
Vol 24 (10) ◽  
pp. 1540006
Author(s):  
Vassily Olegovich Manturov
Keyword(s):  
Group Theory ◽  
Knot Theory ◽  
Free Groups ◽  
Equivalence Classes ◽  
Finitely Presented Groups ◽  
Combinatorial Objects ◽  
Reidemeister Moves ◽  
Interesting Class ◽  
Finitely Presented

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

scholarly journals On subgroups of right angled Artin groups with few generators

2015 ◽  
Vol 25 (04) ◽  
pp. 675-688 ◽  
Author(s):  
Ashot Minasyan
Keyword(s):  
Direct Product ◽  
Free Group ◽  
Free Groups ◽  
Cohomological Dimension ◽  
Artin Group ◽  
Direct Products ◽  
Artin Groups ◽  
Limit Groups ◽  
Right Angled Artin Groups ◽  
Finitely Presented

For each d ∈ ℕ, we construct a 3-generated group Hd, which is a subdirect product of free groups, such that the cohomological dimension of Hd is d. Given a group F and a normal subgroup N ⊳ F we prove that any right angled Artin group containing the special HNN-extension of F with respect to N must also contain F/N. We apply this to construct, for every d ∈ ℕ, a 4-generated group Gd, embeddable into a right angled Artin group, such that the cohomological dimension of Gd is 2 but the cohomological dimension of any right angled Artin group, containing Gd, is at least d. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular, we show that for every n ∈ ℕ there exists δ(n) ∈ ℕ such that any n-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the δ(n)-th direct power of the free group of rank 2. As another corollary we derive that any n-generated finitely presented residually free group embeds into the direct product of at most δ(n) limit groups.

Sulphoamidic Phtalocyaninic Pigments and Dyes(II)

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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