A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS ⊂ G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G → F_Ssuch thath_S(g) ≠ 1 for allg ∈ S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.     

SUBDIRECTLY IRREDUCIBLE LEFT SELF DISTRIBUTIVELY GENERATED ALGEBRAS

Abstract

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R _Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R _Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R _Δ(F) is solvable.     

Daugavet centers and direct sums of Banach spaces

A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X _1⊕F X ₂ of two Banach spaces X ₁ and X ₂ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X ₁ and X ₂ there exists a Daugavet center acting from X _1⊕F X ₂, and the class of those F such that for some pair of spaces X ₁ and X ₂ there is a Daugavet center acting into X _1⊕F X ₂. We also present several examples of such Daugavet centers.     

On division rings generated by polycyclic groups

LetD=F(G) be a division ring generated as a division ring by its central subfieldF and the polycyclic-by-finite subgroupG of its multiplicative group, letn be a positive integer and letX be a finitely generated subgroup of GL(n, D). It is implicit in recent works of A. I. Lichtman thatX is residually finite. In fact, much more is true. If charD=p≠0, then there is a normal subgroup ofX of finite index that is residually a finitep-group. If charD=0, then there exists a cofinite set π=π(X) of rational primes such that for eachp in π there is a normal subgroup ofX of finite index that is residually a finitep-group.     

Real-local fields and the canonical operation of the automorphism group on the space of orderings

The object of our investigation is the canonical operation of the automorphism group of a formally real field F on X_F , the space of orderings of F. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces on X_F the structure of a module over the endomorphism ring of the group of archimedean classes of F. We show that Aut F acts on X_F by affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings of F be transformed into each other by some automorphism of F?"), and to the investigation of the subgroup of Aut F generated by all order automorphism groups of F.     

PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS HAVING 1 AS THE ONLY UNIT

We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F ₂ or to the polynomial ring F ₂[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F ₂ or to F ₂[X]. We also construct principal ideal domains R of infinite transcendence degree over F ₂ with the property that 1 is the only unit of R.

     

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups \mathbb A_m{\mathbb A_m} (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to \mathbb A_m{\mathbb A_m} is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups \mathbb S_m{\mathbb S_m} are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146–182, 1999), and the class of type (2, 3) in \mathbb S₅{\mathbb S_5}. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra \mathfrak B(X, q){\mathfrak B(X, \bf q)} is infinite dimensional, q an arbitrary cocycle.     

Acyclic covers

The following question, which is directly related to the Whitehead problem of subcomplexes of acyclic 2-complexes, is studied: If is a class of groups, X is a 2-dimensional CW-complex and X' is an acyclic, infinite cyclic cover of X with in , must X' be contractible? A positive answer is given if X is finite and is the class of amenable groups. Received: April 24, 1995.     

Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel

Let k be an arbitrary field, X₁,….,X_n indeterminates over k and F₁…, F₃ ε ∈ k[X₁…,X_n] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (F_i). We give an elementary proof of the following effective Nullstellensatz: Assume that F₁,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P₁…, P₃ ε ∈ k[X₁…,X_n] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ P_iF_i and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen-Suslin Theorem baaed on our effective Nullstellensatz.     

ADDITIVE PROPERTIES OF SETS OF REAL NUMBERS AND AN INFINITE GAME

Abstract

1. If X has strong measure zero aid if Y is contained in an F _σ, set of measure zero, then X + Y has measure zero (Proposition 9).

2. If X is a measure zero set with property s ₀ and Y is a Sierpinski set, then X + Y has property s ₀ (Theorem 12).

3. If X is a meager set with property s ₀ and Y is a Lusin set, then X + Y has property s ₀ (Theorem 17).

An infinite game is introduced, motivated by additive properties of certain classes of sets of real numbers.     

Almost Periodic and Pseudo Almost Periodic Mild Solutions to a Partial Differential Equation VIA Fractional Operators

This article is mainly concerned with the existence and uniqueness of almost periodic and pseudo almost periodic mild solutions to a class of partial differential equations in Banach spaces. Under some reasonable assumptions, the main results are proved in α-norm. Our results cover the cases that the functions F ₁; F ₂ take values in different spaces such as X, X _α, and X _ß, where α ≤ β ∈ (0; 1).     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

A criterion for the <Emphasis Type="Italic">F</Emphasis> <Subscript> <Emphasis Type="Italic">π</Emphasis> </Subscript>-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks

Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually F_π, where F_π is the class of all finite p-groups. We prove that G is residually F_π if and only if H is F_π-separable in A and B.     

Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group ${\bigcup_{n\in\mathbb {N}}G_n}We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group è_{n ? \mathbb N}G_n{\bigcup_{n\in\mathbb {N}}G_n} where the G _n are Banach Lie groups.     

Tail asymptotics for the sum of two heavy-tailed dependent risks

Let X ₁ , X ₂ denote positive heavy-tailed random variables with continuous marginal distribution functions F ₁ and F ₂, respectively. The asymptotic behavior of the tail of X ₁ +X ₂ is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F ₁ , F ₂ and the underlying dependence structure of X ₁ and X ₂, we survey explicit asymptotic results available in the literature and add several new cases.Supported by the Austrian Science Fund Project P-18392.     

Homogeneity in relatively free groups

We prove that any torsion-free, residually finite relatively free group of infinite rank is not ${\aleph_1}$ -homogeneous. This generalizes Sklinos’ result that a free group of infinite rank is not ${\aleph_1}$ -homogeneous, and, in particular, gives a new simple proof of that result.     

Residual smallness in commutative algebra

In Oman and Salminen [19 Oman, G., Salminen, A. Residually small commutative rings. J. Commut. Algebra (17 pages as a preprint, to appear). [Google Scholar]], the authors introduce and study residually small rings, defined as follows: an infinite commutative ring R with identity is residually smallif for every r∈R?{0}, there exists an ideal I_r of R such that r?I_r and |R∕I_r|<|R|. The purpose of this note is to extend our study. In particular, we continue our investigation of residually small rings and then generalize this notion to modules.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

A combinatorial condition on a certain variety of groups

Let n be an integer and B_n \mathcal B_n be the variety defined by the law [xⁿ,y][x,yⁿ]^-1 = 1.¶ Let B_n^* \mathcal B_n^* be the class of groups in which for any infinite subsets X, Y there exist x ? X x \in X and y ? Y y \in Y such that [xⁿ,y][x,yⁿ]^-1 = 1. For $ n \in {\pm 2, 3\} $ n \in {\pm 2, 3\} we prove that¶ B_n^* = B_n èF \mathcal B_n^* = \mathcal B_n \cup \mathcal F , F \mathcal F being the class of finite groups. Also for $ n \in {- 3, 4\} $ n \in {- 3, 4\} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ? B_n^* G \in \mathcal B_n^* if and only if G ? B_n G \in \mathcal B_n .