scholarly journals The cup product of Brooks quasimorphisms

2018 ◽  
Vol 30 (5) ◽  
pp. 1157-1162 ◽  
Author(s):  
Michelle Bucher ◽  
Nicolas Monod
Keyword(s):  
Free Group ◽  
Automorphism Groups ◽  
Bounded Cohomology ◽  
Universal Class ◽  
Cup Product

AbstractWe prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Note on a theorem of Magnus

1969 ◽  
Vol 10 (3-4) ◽  
pp. 469-474 ◽  
Author(s):  
Norman Blackburn
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Reduction Theorem ◽  
The South ◽  
Cup Product ◽  
South West ◽  
Image Position ◽  
Independent Parameters

Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.

Centraliser near-rings determined by fixed point free automorphism groups

1987 ◽  
Vol 107 (3-4) ◽  
pp. 327-337 ◽  
Author(s):  
Peter Fuchs ◽  
C. J. Maxson ◽  
M. R. Pettet ◽  
K. C. Smith
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
Nontrivial Ideal

SynopsisLet G be a group and let A be a fixed point free group of automorphisms of G. It is shown that the centraliser near-ring MA(G) has at most one nontrivial ideal. Conditions on the pair (A, G) are given which force MA(G) to be simple. It is shown that if a nonsimple near-ring MA(G) exists, then A and G have unusual properties.

An independence theorem for automorphisms of torsion-free groups

1981 ◽  
Vol 90 (3) ◽  
pp. 403-409
Author(s):  
U. H. M. Webb
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Free Group ◽  
Automorphism Groups ◽  
Torsion Free Group ◽  
Free Nilpotent Group ◽  
Free Abelian Groups ◽  
Independence Theorem ◽  
The Relationship ◽  
Torsion Free

This paper considers the relationship between the automorphism group of a torsion-free nilpotent group and the automorphism groups of its subgroups and factor groups. If G2 is the derived group of the group G let Aut (G, G2) be the group of automorphisms of G which induce the identity on G/G2, and if B is a subgroup of Aut G let B¯ be the image of B in Aut G/Aut (G, G2). A p–group or torsion-free group G is said to be special if G2 coincides with Z(G), the centre of G, and G/G2 and G2 are both elementary abelian p–groups or free abelian groups.

TREE LEVEL LIE ALGEBRA STRUCTURES OF PERTURBATIVE INVARIANTS

2003 ◽  
Vol 12 (03) ◽  
pp. 333-345 ◽  
Author(s):  
NATHAN HABEGGER ◽  
WOLFGANG PITSCH
Keyword(s):  
Lie Algebra ◽  
Free Group ◽  
Automorphism Groups ◽  
Feynman Diagrams ◽  
Tree Level

We study two different Lie algebra structures on the space of Feynman diagrams at tree level. We show that each such structure arises naturally from a tower of automorphism groups of nilpotent quotients of a free group.

scholarly journals KAZHDAN CONSTANTS OF GROUP EXTENSIONS

2010 ◽  
Vol 20 (05) ◽  
pp. 671-688
Author(s):  
UZY HADAD
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Group Extensions ◽  
Abelian Extensions ◽  
Tame Automorphism ◽  
Relatively Free Group

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of p-groups of lower p-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.

On the Andreadakis conjecture restricted to the “lower-triangular” automorphism groups of free groups

2017 ◽  
Vol 16 (05) ◽  
pp. 1750099 ◽  
Author(s):  
Takao Satoh
Keyword(s):  
Cohomology Group ◽  
Homology Group ◽  
Automorphism Groups ◽  
Lower Central Series ◽  
Affirmative Answer ◽  
Central Series ◽  
Homology Groups ◽  
Cup Product ◽  
Lower Triangular ◽  
Cup Products

In this paper, we consider a certain subgroup [Formula: see text] of the IA-automorphism group of a free group. We determine the images of the [Formula: see text]th Johnson homomorphism restricted to [Formula: see text] for any [Formula: see text] and [Formula: see text]. By using this result, we give an affirmative answer to the Andreadakis conjecture restricted for [Formula: see text]. Namely, we show that the intersection of the Andreadakis–Johnson filtration and [Formula: see text] coincides with the lower central series of [Formula: see text]. In a series of this research, we obtain additional results on the integral (co)homology groups of [Formula: see text]. In particular, we determine the first homology group, and study the cup product of first cohomologies of [Formula: see text]. Furthermore, we construct nontrivial second homology classes of [Formula: see text] by observing its generators and relators, and show that the second cohomology group is not generated by cup products of the first cohomology groups.

scholarly journals FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

2014 ◽  
Vol 2 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
SAHARON SHELAH
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Set Theory ◽  
Free Group ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Free Groups ◽  
Large Cardinals ◽  
Free Objects ◽  
Models Of Set Theory

AbstractGiven a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

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