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.

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.

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.

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 Centralisers of Dehn twist automorphisms of free groups

2015 ◽  
Vol 159 (1) ◽  
pp. 89-114 ◽  
Author(s):  
MORITZ RODENHAUSEN ◽  
RICHARD D. WADE
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Dehn Twist ◽  
Classifying Space ◽  
Dehn Twists ◽  
Automorphisms Of Free Groups

AbstractWe refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

Train Tracks and Automorphisms of Free Groups

1992 ◽  
Vol 135 (1) ◽  
pp. 1 ◽  
Author(s):  
Mladen Bestvina ◽  
Michael Handel
Keyword(s):  
Free Groups ◽  
Automorphisms Of Free Groups ◽  
Train Tracks

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.

The Ping-Pong Lemma

2017 ◽  
Author(s):  
Johanna Mangahas
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Group Action ◽  
Metric Space ◽  
Hyperbolic Geometry ◽  
Free Groups ◽  
Central Place ◽  
Geometric Group Theory ◽  
Research Projects ◽  
Ping Pong

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.

Free Groups and Folding

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Membership Problem ◽  
Research Projects ◽  
Residually Finite ◽  
Simple Operation ◽  
Nontrivial Normal Subgroup ◽  
Subgroup Theorem

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.

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.

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