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.

On Basis-Conjugating Automorphisms of Free Groups

1986 ◽  
Vol 38 (6) ◽  
pp. 1525-1529 ◽  
Author(s):  
J. McCool
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Isomorphic Copy ◽  
Generating Set ◽  
Automorphisms Of Free Groups ◽  
Image Position

Let X = {x1, … xn} be a free generating set of the free group Fn and let H be the subgroup of Aut Fn consisting of those automorphisms α such that α(xi) is conjugate to xi for each i = 1, 2 , …, n. We call H the Z-conjugating subgroup of Aut Fn. In [1] Humphries found a generating set for the isomorphic copy H1 of H consisting of Nielsen transformationswhere each is conjugate to ui (see remark 1 below). The purpose of this paper is to find a presentation of H (and hence of H1).Let i ≠ j be elements of {1, 2, …, n}. We denote by (xi; xj) the automorphism of Fn which sends xi to and fixes xk if k ≠ i. Let S be the set of all such automorphisms.

Finite Quotients of the Automorphism Group of a Free Group

1977 ◽  
Vol 29 (3) ◽  
pp. 541-551 ◽  
Author(s):  
Robert Gilman
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Outer Automorphism ◽  
Permutation Representation ◽  
Finitely Generated ◽  
Outer Automorphism Group ◽  
Finite Quotients

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is 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.

scholarly journals Almost congruence extension property for subgroups of free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 125-146
Author(s):  
Lev Glebsky ◽  
Nevarez Nieto Saul
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Free Groups ◽  
Extension Property ◽  
Normal Closure ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Finitely Generated ◽  
Finite Set

AbstractLetHbe a subgroup ofFand{\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}}the normal closure ofHinF. We say thatHhas the Almost Congruence Extension Property (ACEP) inFif there is a finite set of nontrivial elements{\digamma\subset H}such that for any normal subgroupNofHone has{H\cap\langle\kern-1.422638pt\langle N\rangle\kern-1.422638pt\rangle_{F}=N}whenever{N\cap\digamma=\emptyset}. In this paper, we provide a sufficient condition for a subgroup of a free group to not possess ACEP. It also shows that any finitely generated subgroup of a free group satisfies some generalization of ACEP.

scholarly journals Applications of Fox's derivation in determining the generators of a group

2000 ◽  
Vol 61 (1) ◽  
pp. 27-32
Author(s):  
Wan Lin
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Sufficient Conditions ◽  
Inverse Function ◽  
Sufficient Condition ◽  
Inverse Function Theorem ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Necessary And Sufficient

We give a necessary and sufficient condition for a set of elements to be a generating set of a quotient group F/N, where F is the free group of rank n and N is a normal subgroup of F. Birman's Inverse Function Theorem is a corollary of our criterion. As an application of this criterion, we give necessary and sufficient conditions for a set of elements of the Burnside group B (n,p) of exponent p and rank n to be a generating set.

scholarly journals Finite and infinite cyclic extensions of free groups

1973 ◽  
Vol 16 (4) ◽  
pp. 458-466 ◽  
Author(s):  
A. Karrass ◽  
A. Pietrowski ◽  
D. Solitar
Keyword(s):  
Finite Number ◽  
Free Group ◽  
Finite Groups ◽  
Free Groups ◽  
Finite Extension ◽  
Finitely Generated ◽  
Free Products ◽  
Tree Product ◽  
Image Position ◽  
Generalized Free Products

Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

The abelianization of the congruence IA-automorphism group of a free group

2007 ◽  
Vol 142 (2) ◽  
pp. 239-248 ◽  
Author(s):  
TAKAO SATOH
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Quotient Group ◽  
Homology Group ◽  
Integral Homology ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Homology Groups ◽  
Inner Automorphism

AbstractWe consider the abelianizations of some normal subgroups of the automorphism group of a finitely generated free group. Let Fn be a free group of rank n. For d ≥ 2, we consider a group consisting the automorphisms of Fn which act trivially on the first homology group of Fn with ${\mathbf Z}$/d${\mathbf Z}$-coefficients. We call it the congruence IA-automorphism group of level d and denote it by IAn,d. Let IOn,d be the quotient group of the congruence IA-automorphism group of level d by the inner automorphism group of a free group. We determine the abelianization of IAn,d and IOn,d for n ≥ 2 and d ≥ 2. Furthermore, for n=2 and odd prime p, we compute the integral homology groups of IA2,p for any dimension.

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.

Incompressible Surfaces in the Boundary of a Handlebody–an Algorithm

1980 ◽  
Vol 32 (3) ◽  
pp. 590-595 ◽  
Author(s):  
Herbert C. Lyon
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Decomposition Theorem ◽  
Finitely Generated ◽  
Initial Development ◽  
Generating Set ◽  
3 Dimensional ◽  
Incompressible Surfaces ◽  
Minimal Generating Set

Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.

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.

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