Relative cohomology theory for profinite groups

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer–Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components.     

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.     

Automata in Groups and Dynamics and Induced Systems of PDE in Tropical Geometry

In this paper we develop a dynamical scaling limit from rational dynamics to automata in tropical geometry. We compare these dynamics and induce uniform estimates of their orbits. We apply these estimates to introduce a comparison analysis of theory of automata groups in geometric group theory with analysis of rational dynamics and some hyperbolic PDE systems. Frameworks of characteristic properties of automata groups are inherited to the corresponding rational or PDE dynamics. As an application we study the Burnside problem in group theory and translate the property as the infinite quasi-recursiveness in rational dynamics.     

Weakly Linearly Compact Topological Abelian Groups

In this paper, we study linearly topological groups. We introduce the notion of a weakly linearly compact group, which generalizes the notion of a weakly separable group, and examine the main properties of such groups. For weakly linearly compact groups, we construct the character theory and present an algebraic characterization of some classes of such groups. Some well-known theorems for periodic Abelian groups are generalized for the case of linearly discrete, topological Abelian groups; for linearly compact and linearly discrete topological Abelian groups, we also construct the character theory and study some important properties of linearly discrete groups. For linearly discrete, topological Abelian groups, we analyze the splittability condition (Theorem 3.12) and present the characteristic condition of decomposability of a discrete group G into the direct sum of rank-1 groups. We also present an algebraic characterization of linearly compact groups. We introduce the notion of a weakly linearly compact, topological Abelian group, which generalizes the notion of a weakly separable Abelian group, and examine some properties of such groups. These groups are a generalization of fibrous Abelian groups introduced by Vilenkin. We give an algebraic characterization of divisible, weakly locally compact Abelian groups that do not contain nonzero elements of finite order (Proposition 7.9). For weakly locally compact Abelian groups, we construct universal groups.     

Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut-points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well-known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.     

Effective Estimates on Integral Quadratic Forms: Masser’s Conjecture,Generators of Orthogonal Groups,and Bounds in Reduction Theory

In this paper we prove a conjecture of David Masser on small height integral equivalence between integral quadratic forms. Using our resolution of Masser’s conjecture we show that integral orthogonal groups are generated by small elements which is essentially an effective version of Siegel’s theorem on the finite generation of these groups. We also obtain new estimates on reduction theory and representation theory of integral quadratic forms. Our line of attack is to make and exploit the connections between certain problems about quadratic forms and group actions, whence we may study the problem in the well-developed theory of homogeneous dynamics, arithmetic groups, and the spectral theory of automorphic forms.     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

Brauer–Clifford equivalence of full matrix algebras

The Brauer–Clifford group was introduced to describe the Clifford theory for finite groups. It was proved that it has a natural homomorphism into a Brauer group, and the kernel of this homomorphism is the set of all equivalence classes of G-algebras which are full matrix algebras. In this paper, we prove that this kernel is isomorphic to a second cohomology group. In the Clifford theory for finite groups situation, we characterize families of characters which yield elements in the full matrix subgroup of the Brauer–Clifford group as those where an appropriate character has Schur index one. We also show, in this case, how to compute the element of the second cohomology group associated with this family of characters.     

On A-Critical nilpotent groups

It is useful to consider a critical structure of a class of groups. For example, the results on minimal non-nilpotent group, and on minimal non-p-nilpotent group, have been widely used in group theory. In this paper, we analyze the critical structure of a class of groups, which admit an operator group.     

Zero-sum problems in finite abelian groups: A survey

We give an overview of zero-sum theory in finite abelian groups, a subfield of additive group theory and combinatorial number theory. In doing so we concentrate on the algebraic part of the theory and on the development since the appearance of the survey article by Y. Caro in 1996.     

Equivariant Extensions of Categorical Groups

If is a group, then the category of -graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from . In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreiers theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).Mathematics Subject Classifications (2000) 18D10, 18B40, 20J05, 20J06.Partially supported by MTM2004-01060.     

一类2nm(m为奇数)阶有限群的构造

本文研究了一类2nm(m为奇数)阶有限群的构造,利用解数论同余方程的方法和群的扩张理论等知识,得到了具有奇数m阶循环正规子群、其补子群为循环群的2nm阶有限群的构造及相关的计数定理.     

Dimensional dual hyperovals and APN functions with translation groups

In this paper we develop a theory of translation groups for dimensional dual hyperovals and APN functions. It will be seen that both theories can be treated, to a large degree, simultaneously. For small ambient spaces it will be shown that the translation groups are normal in the automorphism group of the respective geometric object. For large ambient spaces there may be more than one translation group. We will determine the structure of the normal closure of the translation groups in the automorphism group and we will exhibit examples which in fact do admit more than one translation group.     

Abelian-by-G Groups,for G Finite,from the Model Theoretic Point of View

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ?[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree. Mathematics Subject Classification: 03C60, 03B25.     

Computing the nonabelian tensor squares of polycyclic groups

In this paper we develop a theory for computing the nonabelian tensor square and related computations for finitely presented groups and specialize it to polycyclic groups. This theory provides a framework for making nonabelian tensor square computations for polycyclic groups and is the basis of an algorithm for computing the nonabelian tensor square for any polycyclic group.     

Powerfully nilpotent groups of maximal powerful class

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.     

Algebraic prolongation and rigidity of Carnot groups

We discuss the known results on rigidity of Carnot groups using Tanaka’s prolongation theory. We also apply Tanaka’s theory to study rigidity of an extended class of H-type groups which we call J-type groups. In particular we obtain a rigidity criterion giving rise to a rigid class of J-type groups which includes the H-type groups, and thus extends the results of H.M. Reimann. We also construct a noncomplex J-type group which is nonrigid and does not satisfy the rank 1 condition over the reals.     

Algebraic prolongation and rigidity of Carnot groups

We discuss the known results on rigidity of Carnot groups using Tanaka’s prolongation theory. We also apply Tanaka’s theory to study rigidity of an extended class of H-type groups which we call J-type groups. In particular we obtain a rigidity criterion giving rise to a rigid class of J-type groups which includes the H-type groups, and thus extends the results of H.M. Reimann. We also construct a noncomplex J-type group which is nonrigid and does not satisfy the rank 1 condition over the reals.     

Apollonian packings and hyperbolic geometry

We review the generalized apollonian packings by Bessis and Demko from 3-dimensional viewpoints and solve their conjectures on the discreteness of the groups they constructed. Moreover, we systematically generalize the construction of packings in terms of the Coxeter group theory, and propose a computational algorithm to draw the pictures efficiently based on the automatic group theory.     

On Turner’s theorem and first-order theory

A theorem of E.C. Turner states that if F is a finitely generated free group, then the test words are precisely the elements not contained in any proper retract. In this paper, we examine some ideas in model theory and logic related to Turner’s characterization of test words and introduce Turner groups, a class of groups containing all finite groups and all stably hyperbolic groups satisfying this characterization. We show that Turner’s theorem is not first-order expressible. However, we prove that every finitely generated elementary free group is a Turner group.