Groups Of Finite Rank Research Articles

Solvable quotients of Kähler groups

AbstractWe prove several results on the structure of solvable quotients of funda-mental groups of compact Ka¨hler manifolds (Ka¨hler groups). 1. Introduction. We first recall a definition from [AN].Definition 1.1 A solvable group Γ has finite rank, if there is a decreasing sequenceΓ = Γ 0 ⊃ Γ 1 ⊃ ... ⊃ Γ m+1 = {1} of subgroups, each normal in its predecessor, suchthat Γ i /Γ i+1 is abelian and Q ⊗(Γ i /Γ i+1 ) is finite dimensional for all i.In what follows F r denotes a free group with the number of generators r ∈ Z + ∪{∞}.Our main result isTheorem 1.2 Let M be a compact Kahler manifold. Assume that the fundamentalgroup π 1 (M) is defined by the sequence{1} −→ F −→ π 1 (M)−→ p H −→ {1}where H is a solvable group of finite rank of the form{0} −→ A −→ H −→ B −→ {0}with non-trivial abelian groups A,B so that Q ⊗ A ∼= Q m and m ≥ 1. Assume alsothat p −1 (A) ⊂ π 1 (M) does not admit a surjective homomorphism onto F ∞ . Then alleigen-characters of the conjugate action of B on the vector space Q⊗A are torsion.In Lemma 2.3 we will show that the condition for p

Induced Representations of Abelian Groups of Finite Rank

We prove that any irreducible faithful representation of an almost torsion-free Abelian group G of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group G.

Automorphisms of nearly finite Coxeter groups

Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n � 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðW Þ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If, in addition, WJ is irreducible and every conjugacy class of reflections in W has nonempty intersection with WJ , then all automorphisms of W preserve reflections, and it follows that AutðW Þ is the semi-direct product of InnðW Þ by the group of graph automorphisms.

Automorphic orbits in free groups

Let F n be the free group of a finite rank n. We study orbits Orb φ ( u), where u is an element of the group F n , under the action of an automorphism φ. If an orbit like that is finite, we determine precisely what its cardinality can be if u runs through the whole group F n , and φ runs through the whole group Aut( F n ). Another problem that we address here is related to Whitehead's algorithm that determines whether or not a given element of a free group of finite rank is an automorphic image of another given element. It is known that the first part of this algorithm (reducing a given free word to a free word of minimum possible length by elementary Whitehead automorphisms) is fast (of quadratic time with respect to the length of the word). On the other hand, the second part of the algorithm (applied to two words of the same minimum length) was always considered very slow. We give here an improved algorithm for the second part, and we believe this algorithm always terminates in polynomial time with respect to the length of the words. We prove that this is indeed the case if the free group has rank 2.

A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group

AbstractWe show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

The Fixed Point Group of a Free Group Automorphism

It is proved that the fixed point group of an arbitrary automorphism of a free group of finite rank has an algorithmically computable basis.

Wreath operations in the group of automorphisms of the binary tree

A new operation called tree-wreathing is defined on groups of automorphisms of the binary tree. Given a countable residually finite 2-group H and a free abelian group K of finite rank r this operation produces uniformly copies of these as automorphism groups of the binary tree such that the group generated by them is an over-group of the restricted wreath product H≀K . Indeed, G contains a normal subgroup N which is an infinite direct sum of copies of the derived group H ′ and the quotient group G / N is isomorphic to H≀K . The tree-wreathing construction preserves the properties of solvability, torsion-freeness and of having finite state (i.e., generated by finite automata). A faithful representation of any free metabelian group of finite rank is obtained as a finite-state group of automorphisms of the binary tree.

Maximal index automorphisms of free groups with no attracting fixed points on the boundary are Dehn twists

In this paper we define a quantity called the rank of an outer automorphism of a free group which is the same as the index introduced in [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, `An index for counting fixed points for automorphisms of free groups', Duke Math. J. 93 (1998) 425-452] without the count of fixed points on the boundary. We proceed to analyze outer automorphisms of maximal rank and obtain results analogous to those in [D.J. Collins and E. Turner, `An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element', J. Pure and Applied Algebra 88 (1993) 43-49]. We also deduce that all such outer automorphisms can be represented by Dehn twists, thus proving the converse to a result in [M.M. Cohen and M. Lustig, `The conjugacy problem for Dehn twist automorphisms of free groups', Comment Math. Helv. 74 (1999) 179-200], and indicate a solution to the conjugacy problem when such automorphisms are given in terms of images of a basis, thus providing a moderate extension to the main theorem of Cohen and Lustig by somewhat different methods.

The classification problem for torsion-free abelian groups of finite rank

We prove that for each n ≥ 1 n \geq 1 , the classification problem for torsion-free abelian groups of rank n + 1 n+1 is not Borel reducible to that for torsion-free abelian groups of rank n n .

CLASSIFICATION AND DIRECT DECOMPOSITIONS OF SOME BUTLER GROUPS OF COUNTABLE RANK

ABSTRACT This paper is devoted to an extension of results concerning decompositions of particular almost completely decomposable groups of finite rank studied by Mader and the first author in 1-2 to groups of infinite rank. Thus we obtain asystematic way to consider all decompositions of some fairly large class of torsion-free abelian groups. This will be illustrated discussing all pathological decompositions of Corner's[3] group constructed 40 years ago.

Just infinite modules over metabelian groups of finite rank

It is proved, in particular, that if is a metabelian group of finite rank and is a faithful just infinite -module, then is finitely generated. This includes studying properties of induced modules over the group algebra of a metabelian group of finite rank over a field of arbitrary characteristic.

ON PROFINITE GROUPS WHOSE POWER SUBGROUPS ARE CLOSED

ABSTRACT We say that a profinite group G has Property S if for each integer n, there is a bound k such that every element of the nth power subgroup is a product of k nth powers of elements of G. Finitely generated profinite groups with Property S are completely determined by their group structure in a way conjectured by Hartley to be true of all finitely generated profinite groups; namely, every subgroup of finite index is open. We show that the class of finitely generated profinite groups with Property S is closed under forming extensions of its members and under taking subgroups of finite index. As a consequence, we note that all profinite groups of finite rank have Property S.

On Solvable Groups with Proper Quotient Groups of Finite Rank

We study solvable groups of infinite special rank all proper normal subgroups of which define quotient groups of finite special rank.

SECOND ORDER DEHN FUNCTIONS OF SPLIT EXTENSIONS OF THE FORM

ABSTRACT The purpose of this paper to show the upper and lower bounds of the second order Dehn functions of particular split extensions of the form , where F is the free group of finite rank freely generated by a finite set . To obtain these bounds we use some geometric techniques by constructing the generating spherical pictures of the second homotopy module of the relevant group presentations.

On Primitive Representations of Minimax Nilpotent Groups

We show, in particular, that in the class of minimax two-step nilpotent groups only finitely generated groups can admit exact irreducible primitive representations over a finitely generated field of characteristic zero. We also suggest some approaches to studying induced representations of nilpotent groups of finite rank.

Computation of the Mapping of Factorization by the Radical for K 0 + of the Endomorphism Ring

The mapping of factorization by the radical is computed for the semigroup of projective, finitely generated modules over the endomorphism ring of an almost completely decomposable torsion-free Abelian group of finite rank that is divisible by almost all prime numbers. Also, an answer is given to the question concerning the collections of groups of rank 1 for which one can construct an almost completely decomposable group, indecomposable as an object in \(\bar M^p\), by adding a generator. Bibliography: 4 titles.

Self-Cancellation of Torsion-Free Abelian Groups of Finite Rank

A very simple example of an Abelian group free of self-cancellation is constructed. Bibliography: 11 titles.

On the Krull―Schmidt Theorem for Artinian Modules

A property of rings generalizing commutativity is introduced. If a ring satisfies this property, then the Krull--Schmidt theorem holds for Artinian modules over the ring. In particular, this property is fulfilled for local rings of finite rank and for rings such that their centers are surjectively mapped by the natural projection onto the factor with respect to the radical of the ring. A local ring for which the property fails is constructed; for the direct decompositions of Artinian modules over this ring there appear anomalies similar to the anomalies of direct decompositions of torsion-free Abelian groups of finite rank. Bibliography: 6 titles.

POWER GROUPS OF FREE ABELIAN GROUPS OF FINITE RANK

ABSTRACT The power set of a group G has an induced semigroup structure, some subsets of which will form groups in their own right. We are especially interested in such subsets that are maximal. We demonstrate that even when G is a free abelian group of finite rank, the groups which arise in this way can be diverse profinite abelian groups.

Direct sums of local torsion-free abelian groups

The category of local torsion-free abelian groups of finite rank is known to have the cancellation and n n -th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960’s.