Non-abelian Free Subgroup Research Articles

An example of an end-type action on a Λ-tree

Let Λ be an ordered abelian group. We give an example of a non-trivial action of end type of a group G on a Λ-tree which is faithful and transitive and such that G is not soluble and contains no non-abelian free subgroup. The degree of each point is 3. This is related to certain automorphisms of the infinite rooted binary tree.

The Tits alternative for generalized tetrahedron groups

A generalised tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalised triangle groups and whose edge groups are finite cyclic. We prove an improved Spelling Theorem for generalised triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalised tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a nonabelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalised tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a nonabelian free subgroup.

Normal subgroups generated by a single pure element in quaternion algebras

Let D be a quaternion division algebra whose center is an arbitrary infinite field K of characteristic ≠2, and let e ∈ D be a pure quaternion. Hence, by definition, e ∈ D ∖ K and e 2 ∈ K . We show that if the characteristic of K is >2, then D × / 〈 e D × 〉 is abelian-by-nilpotent-by-abelian. Note that by [A.S. Rapinchuk, L. Rowen, Y. Segev, Nonabelian free subgroups in homomorphic images of valued quaternion division algebras, Proc. Amer. Math. Soc., in press] this result is false in characteristic zero. As a consequence we show that the Whitehead group W ( G , k ) , where G is an absolutely simple simply connected algebraic group of type D 4 3 , 6 defined over a field k of odd characteristic and of k-rank 1, is abelian-by-nilpotent-by-abelian.

EXTENDED FINITE AUTOMATA AND WORD PROBLEMS

This paper considers extended finite automata over monoids, in the sense of Dassow and Mitrana. We show that the family of languages accepted by extended finite automata over a monoid K is controlled by the word problem of K in a precisely stated manner. We also point out a critical error in the proof of the main result in the paper by Dassow and Mitrana. However as one consequence of our approach, by analyzing a certain word problem, we obtain a complete proof of this result, namely that the family of languages accepted by extended finite automata over the free group of rank two is exactly the family of context-free languages. We further deduce that along with the free group of rank two, the only finitely generated groups with this property are precisely the groups that have a nonabelian free subgroup of finite index.

Coxeter covers of the symmetric groups

We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set T of transpositions. These quotients, denoted here by C Y ( T ), are a special case of the generalized Coxeter groups defined in [5], and also arise in the computation of certain invariants of surfaces. We use a surprising action of S n on the kernel of the surjection C Y ( T ) → S n to show that this kernel embeds in the direct product of n copies of the free group π 1 ( T ), except when T is the full set of transpositions in S 4 . As a result, we show that each group C Y ( T ) either is virtually Abelian or contains a non-Abelian free subgroup.

Characteristic subsurfaces and Dehn filling

Let M M be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in M M which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of M ( α ) M(\alpha ) has no non-abelian free subgroup, and if M ( β ) M(\beta ) is a reducible manifold which is not homeomorphic to S 1 × S 2 S^1 \times S^2 or P 3 # P 3 P^3 \# P^3 , then Δ ( α , β ) ≤ 5 \Delta (\alpha , \beta )\le 5 . Under the same condition on M ( β ) M(\beta ) , it is shown that if M ( α ) M(\alpha ) is Seifert fibered, then Δ ( α , β ) ≤ 6 \Delta (\alpha , \beta )\le 6 . Moreover, in the latter situation, character variety techniques are used to characterize the topological types of M ( α ) M(\alpha ) and M ( β ) M(\beta ) in case the bound of 6 6 is attained.

WHEN IS THE AUTOMORPHISM GROUP OF A VIRTUALLY POLYCYCLIC GROUP VIRTUALLY POLYCYCLIC?

The automorphism group of a virtually polycyclic group $G$ is either virtually polycyclic or it contains a non-abelian free subgroup. We describe conditions on the structure of $G$ to decide which of the two alternatives occurs for $Aut(G).$

On Units of Twisted Group Algebras

We study units of twisted group algebras. Let G be a finite group and K be a field of characteristic p > 0. Assume that K is not algebraic over a finite field. We determine when units of Kt[G] do not contain any nonabelian free subgroup. We also discuss what will happen when G is locally finite. For twisted group algebras of locally finite groups over any infinite field of characteristic p > 0, we characterize those twisted group algebras with units satisfying a group identity. Finally, we include a new characterization for twisted group algebras to satisfy a polynomial identity.

Left ordered groups with no non-abelian free subgroups

Left ordered groups with no non-abelian free subgroups

A tits alternative for groups that are residually of bounded rank

In this short note a Tits alternative for certain kinds of groups which are residually of rank at mostr is obtained. The main theorem states that ifG is a group that is residually (locally (soluble-by-finite) of rankr), then eitherG is locally (soluble-by-finite) orG contains a non-abelian free subgroup.

Algorithms for Matrix Groups and the Tits Alternative

Tits has shown that a finitely generated matrix group either contains a nonabelian free group or has a solvable subgroup of finite index. We give a polynomial time algorithm for deciding which of these two conditions holds for a given finitely generated matrix group over an algebraic number field. Noting that many computational problems are undecidable for groups with nonabelian free subgroups, we investigate the complexity of problems relating to matrix groups with solvable subgroups of finite index. For such a groupG, we are able in polynomial time to compute a homomorphismφsuch thatφ(G) is a finite matrix group and the kernel ofφis solvable. If, in addition,Ghas a nilpotent subgroup of finite index, we obtain much stronger results. For such groups, we show how to effectively compute an encoding of elements ofGsuch that the encoding length of an element obtained as a product of length ⩽ℓ over the generators isO(logℓ) times a polynomial in the input length. This result is the best possible; it has been shown by Tits and Wolf that if a finitely generated matrix group does not have a nilpotent subgroup of finite index, then the number of group elements expressible as words of length ℓ over the generators grows ascℓfor some constantc>1 depending onG. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups.

Subnormal subgroups of group ring units

Let G G be an arbitrary group. If a ∈ Z G a\in \mathbb {Z}G satisfies a 2 = 0 a^{2}=0 , a ≠ 0 a\ne 0 , then the units 1 + a 1+a , 1 + a ∗ 1+a^{*} generate a nonabelian free subgroup of units. As an application we show that if G G is contained in an almost subnormal subgroup V V of units in Z G \mathbb {Z}G then either V V contains a nonabelian free subgroup or all finite subgroups of G G are normal. This was known before to be true for finite groups G G only.

Constructing free subgroups of integral group ring units

Let G G be an arbitrary group. It is proved that if Z G \mathbb {Z}G contains a bicyclic unit u ≠ 1 u\ne 1 , then ⟨ u , u ∗ ⟩ \langle u,u^*\rangle is a nonabelian free subgroup of invertible elements.

Actions Propres sur les Espaces Homogenes Reductifs

Let G be a linear semisimple real Lie group and H be a re- ductive subgroup of G. We give a necessary and sufficient condition for the existence of a nonabelian free discrete subgroup r of G acting properly on G/H. For instance, such a group r does exist for SL(2n, IR)/SL(2n - 1, IR) but does not for SL(2n + 1, R)/SL(2n, R) with n > 1.

SUBGROUPS SATISFYING AN IDENTITY IN A CLASS OF ABSTRACT GROUPS

A description is obtained of the subgroups of groups acting on a tree that do not contain nonabelian free subgroups; it is a new interpretation of a result of Bass. The author considers the class consisting of all groups constructive from cyclic groups using amalgamated free products and HNN-extensions, with certain restrictions. A description is obtained of all the subgroups of groups in that satisfy identities, and it is shown that the groups in satisfy the Tits alternative. The proof uses the techniques of group actions on trees.

Discrete groups and simple C*-algebras

Let G denote a discrete group and let us say that G is C*-simple if the reduced group C*-algebra associated with G is simple. We notice immediately that there is no interest in considering here the full group C*-algebra associated with G, because it is simple if and only if C is trivial. Since Powers in 1975 [26] proved that all nonabelian free groups are C*-simple, the class of C*-simp1e groups has been considerably enlarged (see [1, 2, 6, 7, 12, 13, 14, 16, 24] as a sample!), and two important subclasses are so-called weak Powers groups ([6, 13]; see Section 4 for definition and examples) and the groups of Akemann-Lee type [1, 2], which are groups possessing a normal non-abelian free subgroup with trivial centralizer.

Fundamental groups of manifolds of nonpositive curvature

Introduction The universal covering space H of a complete Riemannian manifold M of nonpositive sectional curvature is diffeomorphic to R, n — dim(Λf). Hence the homotopy type of M is completely determined by the isomorphism class of the fundamental group Γ of M. It is, therefore, only natural to expect strong relations between the geometric structure of M and the algebraic structure of Γ. In this paper we obtain several such relations: A general assumption in the results we state below is that (1) the sectional curvature is nonpositive and bounded from below by some constant -a and (2) the volume of M is finite. We define the rank of a unit tangent vector υ of M, rank(u), to be the dimension of the space of all parallel Jacobi fields along the geodesic yv which has initial velocity v. The minimum of rank(ί ) over all v e SM is called the rank of M. This agrees with the usual rank if M is a locally symmetric space. Manifolds of rank one resemble manifolds of strictly negative curvature (see [2] [3], and §2 below). Manifolds of higher rank are studied in [5], [6], [7], and [3], and the conclusive result is that H is a space of rank one, or a symmetric space, or a Riemannian product of such spaces. This is the basic ingredient in the proofs of our results; it allows us, more or less, to consider only the cases that H is of rank one or a symmetric space. As a first example of this principle we indicate in our preliminary section the proof of the following theorem. Theorem A. Either M is flat or Γ contains a nonabelian free subgroup. This is an improvement of the result of Avez [1] that Γ has exponential growth if M is compact and not flat.

Translation invariant functionals on Lp (G) when G is not amenable

AbstractIt is shown that if G is a non-amenable group, then there are no non-zero translation invariant functionals on Lp(G) for 1 < p < ∞. Furthermore, if G contains a closed, non-abelian free subgroup, then there are no non-zero translation invariant functionals on C0(G). The latter is proved by showing that a certain non-invertible convolution operator on C0(G) is surjective.

NORMAL SUBGROUPS OF FREE CONSTRUCTIONS

In this paper the technique of group action on a tree is used to obtain solutions of the following problems. Suppose that the group is a free construction.1. Describe the normal subgroups of not containing nonabelian free subgroups.2. Describe the normal subgroups and of if the mutual commutator subgroup does not contain nonabelian free subgroups.The results are applied to groups obtained by using a sequence of operations of taking HNN-extensions and forming free products with amalgamation.Bibliography: 16 titles.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.