scholarly journals Local structure of some Out(Fn)-complexes

1990 ◽  
Vol 33 (3) ◽  
pp. 367-379 ◽  
Author(s):  
Karen Vogtmann
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Local Structure ◽  
Simplicial Complexes ◽  
Homotopy Equivalent ◽  
Outer Automorphisms ◽  
Finite Quotient

In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1.

scholarly journals On the generic type of the free group

2011 ◽  
Vol 76 (1) ◽  
pp. 227-234 ◽  
Author(s):  
Rizos Sklinos
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Primitive Elements ◽  
Generic Type

AbstractWe answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed in Fω. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not ℵ1-homogeneous.

scholarly journals A note on groups with separable finitely generated subgroups

1987 ◽  
Vol 36 (1) ◽  
pp. 153-160 ◽  
Author(s):  
R. G. Burns ◽  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Cyclic Extension ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Positive Direction ◽  
One Relator Groups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

scholarly journals ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1611-1634 ◽  
Author(s):  
ABDÓ ROIG ◽  
ENRIC VENTURA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Finite Rank ◽  
Polynomial Algorithm ◽  
Minimum Size ◽  
Input Word ◽  
Finitely Generated ◽  
Factor Problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved 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.

Out(F3) Index Realization

2015 ◽  
Vol 159 (3) ◽  
pp. 445-458 ◽  
Author(s):  
CATHERINE PFAFF
Keyword(s):  
Free Group ◽  
Quadratic Differentials ◽  
Singularity Index ◽  
Outer Automorphisms ◽  
Teichmüller Flow ◽  
Surface Homeomorphism

AbstractBy proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this paper we determine an analog to the theorem forOut(F3). That is, we determine which index lists permitted by the [GJLL98] index sum inequality are achieved by ageometric fully irreducible outer automorphisms of the rank-3 free group.

scholarly journals Proficient presentations and direct products of finite groups

1999 ◽  
Vol 60 (2) ◽  
pp. 177-189 ◽  
Author(s):  
K.W. Gruenberg ◽  
L.G. Kovács
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Module Structure ◽  
Direct Products ◽  
The Difference ◽  
A Finite Group ◽  
Open Question

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.

scholarly journals Generating groups of certain soluble varieties

1974 ◽  
Vol 17 (2) ◽  
pp. 222-233 ◽  
Author(s):  
Narain Gupta ◽  
Frank Levin
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Infinite Rank ◽  
Variety Of Groups ◽  
Countably Infinite

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)

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