On the Universal SL2-Representation Rings of Free Groups

2017 ◽  
Vol 60 (4) ◽  
pp. 973-1001
Author(s):  
Takao Satoh
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Free Groups ◽  
Natural Action ◽  
Representation Rings ◽  
Explicit Realization

AbstractIn this paper, we give an explicit realization of the universal SL2-representation rings of free groups by using ‘the ring of component functions’ of SL(2, ℂ)-representations of free groups. We introduce a descending filtration of the ring, and determine the structure of its graded quotients. Then we study the natural action of the automorphism group of a free group on the graded quotients, and introduce a generalized Johnson homomorphism. In the latter part of this paper, we investigate some properties of these homomorphisms from a viewpoint of twisted cohomologies of the automorphism group of a free group.

scholarly journals COMPRESSED WORDS AND AUTOMORPHISMS IN FULLY RESIDUALLY FREE GROUPS

2010 ◽  
Vol 20 (03) ◽  
pp. 343-355 ◽  
Author(s):  
JEREMY MACDONALD
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Free Group ◽  
Polynomial Time ◽  
Free Groups ◽  
Finitely Generated

We show that the compressed word problem in a finitely generated fully residually free group ([Formula: see text]-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an [Formula: see text]-group is decidable in polynomial time.

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

scholarly journals On Commutator Equalities and Stabilizers in Free Groups

1976 ◽  
Vol 19 (3) ◽  
pp. 263-267 ◽  
Author(s):  
R. G. Burns ◽  
C. C. Edmunds ◽  
I. H. Farouqi
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Simple Proof ◽  
Free Groups

AbstractA simple proof is given of a result of Hmelevskiï on the solutions of the equation [x, y] = [u, υ] over a free group for any specified u, υ. To illustrate, the equation is solved explicitly for (u, υ) = (a, b), (a2, b), ([a, b], c) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commutators in the automorphism group of this free group are determined.

Some Remarks on IA Automorphisms of Free Groups

1988 ◽  
Vol 40 (5) ◽  
pp. 1144-1155 ◽  
Author(s):  
J. McCool
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Generating Set ◽  
Automorphisms Of Free Groups ◽  
Identity Automorphism ◽  
Image Position

Let An be the automorphism group of the free group Fn of rank n, and let Kn be the normal subgroup of An consisting of those elements which induce the identity automorphism in the commutator quotient group . The group Kn has been called the group of IA automorphisms of Fn (see e.g. [1]). It was shown by Magnus [7] using earlier work of Nielsen [11] that Kn is finitely generated, with generating set the automorphismsandwhere x1, x2, …, xn, is a chosen basis of Fn.

Automorphisms of Free Groups

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Free Group ◽  
Free Groups ◽  
Research Projects ◽  
Frobenius Theorem ◽  
Automorphisms Of Free Groups ◽  
Train Tracks

This chapter discusses the automorphisms of free groups. Every group is the collection of symmetries of some object, namely, its Cayley graph. A symmetry of a group is called an automorphism; it is merely an isomorphism of the group to itself. The collection of all of the automorphisms is also a group too, known as the automorphism group and denoted by Aut (G). The chapter considers basic examples of groups to illustrate what an automorphism is, with a focus on the automorphisms of the symmetric group on three elements and of the free abelian group. It also examines the dynamics of an automorphism of a free group and concludes with a description of train tracks, a topological model for the free group, and the Perron–Frobenius theorem. Exercises and research projects are included.

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.

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

Subgroups of Finite Index in Free Groups

1949 ◽  
Vol 1 (2) ◽  
pp. 187-190 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Recursion Formula ◽  
Independent Interest ◽  
Finiteness Conditions ◽  
Total Length ◽  
Free Generators

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

scholarly journals A dendrological proof of the Scott conjecture for automorphisms of free groups

1998 ◽  
Vol 41 (2) ◽  
pp. 325-332 ◽  
Author(s):  
D. Gaboriau ◽  
G. Levitt ◽  
M. Lustig
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Alternative Proof ◽  
Automorphisms Of Free Groups

Let α be an automorphism of a free group of rank n. The Scott conjecture, proved by Bestvina-Handel, asserts that the fixed subgroup of α has rank at most n. We give a short alternative proof of this result using R-trees.

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

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