Bases and Automorphisms of the Free Group on Two Generators

2018 ◽  
pp. 125-142
Author(s):  
Christophe Reutenauer
Keyword(s):  
Free Group ◽  
Efficient Algorithm ◽  
Group Of Automorphisms ◽  
Inner Automorphism

The chapter begins with a self-contained exposition of the theory of Nielsen on the free groupwith two generators: bases of F(a, b),Nielsen’s criterion for automorphisms of F(a, b), It also coversNielsen’s theoremon abelianization of these automorphisms andWeinbaum’s theorem on representatives of the group of automorphisms modulo the subgroup of inner automorphism. Perrine’s theorem on bases of the derived group of SL2(Z) and Markoff triples is deduced, and a very simple and efficient algorithm for detecting bases of F(a, b) is given (Séébold, Kassel, the author). Positive automorphisms of F(a, b) are characterized (Wen andWen) and shown to coincide with Sturmian morphisms (Mignosi, Séébold).

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.

scholarly journals AUTOMORPHIC ORBITS IN FREE GROUPS: WORDS VERSUS SUBGROUPS

2010 ◽  
Vol 20 (04) ◽  
pp. 561-590 ◽  
Author(s):  
PEDRO V. SILVA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Group Of Automorphisms ◽  
Rational Subset ◽  
Rank 2 ◽  
Higher Rank

We show that the following problems are decidable in a rank 2 free group F2: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? Moreover, decidability subsists if we allow H to be a rational subset of F2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

An efficient algorithm for constructing the basis of a subgroup of a free group

Cybernetics ◽  
1982 ◽  
Vol 17 (3) ◽  
pp. 407-416 ◽  
Author(s):  
A. A. Letichevskii ◽  
A. B. Godlevskii ◽  
S. L. Krivoi
Keyword(s):  
Free Group ◽  
Efficient Algorithm

A faithful polynomial representation of Out F3

1989 ◽  
Vol 106 (2) ◽  
pp. 207-213 ◽  
Author(s):  
James McCool
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Polynomial Ring ◽  
Outer Automorphism ◽  
Faithful Representation ◽  
Partial Answer ◽  
Polynomial Representation ◽  
Group Of Automorphisms ◽  
Outer Automorphism Group ◽  
Torsion Free

Let Fn be a free group of rank n and let Out Fn be its outer automorphism group. The main result of this paper is that Out F3 has a faithful representation as a group of automorphisms of the polynomial ring in seven variables over the integers. This extends a similar result for n = 2 (see Helling [3], Horowitz [5] and Rosenberger [12]), and provides a partial answer to a conjecture attributed in [5] to W. Magnus. As an application of the special nature of the representing polynomials, we obtain our second result, that Out F3 is virtually residually torsion-free nilpotent.

Nielsen's commutator test for two-generator groups

1993 ◽  
Vol 114 (2) ◽  
pp. 295-301 ◽  
Author(s):  
Narain Gupta ◽  
Vladimir Shpilrain
Keyword(s):  
Free Group ◽  
Final Section ◽  
Commutator Subgroup ◽  
Metabelian Group ◽  
Free Metabelian Group ◽  
Sufficiency Condition ◽  
Inner Automorphism ◽  
Tame Automorphism ◽  
Polynilpotent Group ◽  
Inner Automorphisms

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

scholarly journals THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS

2015 ◽  
Vol 58 (1) ◽  
pp. 153-176 ◽  
Author(s):  
MICHAEL BRANDENBURSKY ◽  
ŚWIATOSŁAW R. GAL ◽  
JAREK KĘDRA ◽  
MICHAŁ MARCINKOWSKI
Keyword(s):  
Cayley Graph ◽  
Free Group ◽  
Efficient Algorithm ◽  
Isometric Embedding ◽  
Finitely Generated ◽  
Locally Compact ◽  
Cyclic Subgroups ◽  
Geometric Origin ◽  
Word Norm

AbstractWe study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

scholarly journals POLYNOMIAL-TIME COMPLEXITY FOR INSTANCES OF THE ENDOMORPHISM PROBLEM IN FREE GROUPS

2007 ◽  
Vol 17 (02) ◽  
pp. 289-328 ◽  
Author(s):  
LAURA CIOBANU
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Efficient Algorithm ◽  
Time Complexity ◽  
Free Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Polynomial Time Complexity

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism ϕ of F sending W to U. This work analyzes an approach due to Edmunds and improved by Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two-generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

scholarly journals RANDOM GENERATION OF FINITELY GENERATED SUBGROUPS OF A FREE GROUP

2008 ◽  
Vol 18 (02) ◽  
pp. 375-405 ◽  
Author(s):  
FRÉDÉRIQUE BASSINO ◽  
CYRIL NICAUD ◽  
PASCAL WEIL
Keyword(s):  
Uniform Distribution ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Efficient Algorithm ◽  
Random Generation ◽  
Finitely Generated ◽  
Average Case ◽  
Case Complexity ◽  
Average Case Complexity

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity [Formula: see text] in the RAM model and [Formula: see text] in the bitcost model.

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