The Euler Class Group of a Noetherian Ring

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dim A and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.     

Approximability of finite rank soluble groups by certain classes of finite groups

For soluble groups of finite rank we obtain the necessary and sufficient condition to be a virtually residually finite p-group. We also prove that a soluble group G of finite rank is residually π-finite for some finite set π of primes if and only if it has no subgroups of type Q and the torsion radical of G is a finite group.     

Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups

We consider a representation of quasi-endomorphisms of Abelian torsion-free groups of rank 4 bymatrices of order 4 over the field of rational numbers Q. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank 4 quasi-decomposable into a direct sum of groups A ₁, A ₂ of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ? Hom(A _i , B) and Q ? Hom(B, A _i ) for any i = 1, 2 have rank 1 or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.     

Stability results for projective modules over Rees algebras

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R-module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.     

Projective modules over the real algebraic sphere of dimension 3

Let A be a commutative noetherian ring of Krull dimension 3. We give a necessary and sufficient condition for A-projective modules of rank 2 to be free. Using this, we show that all the finitely generated projective modules over the algebraic real 3-sphere are free.     

Assouad-Nagata dimension of locally finite groups and asymptotic cones

In this paper we study two problems concerning Assouad-Nagata dimension:

(1)

Is there a metric space of positive asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]).

(2)

Suppose G is a locally finite group with a proper left invariant metric dG. If dimAN(G,dG)>0, is dimAN(G,dG) infinite? (Brodskiy et al., preprint [6, Problem 5.3]).

The first question is answered positively. We provide examples of metric spaces of positive (even infinite) Assouad-Nagata dimension such that all of its asymptotic cones are ultrametric. The metric spaces can be groups with proper left invariant metrics.The second question has a negative solution. We show that for each n there exists a locally finite group of Assouad-Nagata dimension n. As a consequence this solves for non-finitely generated countable groups the question about the existence of metric spaces of finite asymptotic dimension whose asymptotic Assouad-Nagata dimension is larger but finite.     

Intersection theory of algebraic obstructions

Let A be a noetherian commutative ring of dimension d and L be a rank one projectiveA-module. For 1≤r≤d, we define obstruction groups E^r(A,L). This extends the original definition due to Nori, in the case r=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) E^r(A,A)×E^s(A,A)→E^r+s(A,A). For a projective A-module Q of rank n≤d, with an orientation , we define a Chern class like homomorphism

w(Q,χ):E^d−n(A,L^′)→E^d(A,LL^′),     

Nonsingular solutions of TA−BT=C

When A, B and C are given square matrices and C is of rank one, sufficient conditions are given for every solution to be nonsingular when solutions exist. When C has arbitrary rank, some sufficient conditions are given; and when, additionally, A and B have disjoint spectra, necessary conditions are given.     

Left division in the free left distributive algebra on one generator

Let A be the free algebra on one generator satisfying the left distributive law a(bc)=(ab)(ac). Using a division algorithm for elements of an extension P of A, we prove some facts about left division in A, one consequence of which is a conjecture of J. Moody: If a,b,c,d∈A,ab=cd,a and b have no common left divisors, and c and d have no common left divisors, then a=c and b=d.     

Two-parameter families of quantum symmetry groups

We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O⁺(p,q), B⁺(p,q), S⁺(p,q) and H⁺(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier. For H⁺(p,q) the situation is different and we show that , where the latter can be viewed as a liberation of the classical isometry group of the p-dimensional torus.     

On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and ^∗〈A^∗A〉 be the quotient Banach algebra of A∗∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of ^∗〈A^∗A〉 but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Zt(^∗〈A^∗A〉) of ^∗〈A^∗A〉 are obtained, and a characterization of Zt(^∗〈A^∗A〉) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Zt(^∗〈A^∗A〉) are also answered.     

Finite groups with some CAP-subgroups

Let G be a finite group. A subgroup H of G is called a CAP-subgroup if the following condition is satisfied: for each chief factor K/L of G either HK = HL or H ∩ K = H ∩ L. Let p be a prime factor of |G| and let P be a Sylow p-subgroup of G. If d is the minimum number of generators of P then there exists a family of maximal subgroups of P, denoted by M _d (P)={P ₁, P ₂,…, P _d } such that ∩ _i=1 ^d P _i = ?(P). In this paper, we investigate the group G satisfying the condition: every member of a fixed M _d (P) is a CAP-subgroup of G. For example, if, in addition, G is p-solvable, then G is p-supersolvable.     

Primitivity in the free groups of the variety UmUn

For each m,n >= 0, let G_m,n denote the free group of rank r in the variety U_mU_n. The main results in this paper are: (i) a necessary and sufficient condition for a system of r elements {v₁,…, v_r} to form a basis of G_m,ni (ii) necessary and sufficient conditions for a system of l elements {v₁,…, v_l}, l <= r, to be included in a basis of G_m,0. In particular, (i), (ii) yield corresponding results for the free metabelian group of rank r.     

Sum formulas for reductive algebraic groups

Let V be a Weyl module either for a reductive algebraic group G or for the corresponding quantum group Uq. If G is defined over a field of positive characteristic p, respectively if q is a primitive lth root of unity (in an arbitrary field) then V has a Jantzen filtration V=V⁰⊃V¹⊃?⊃Vr=0. The sum of the positive terms in this filtration satisfies a well-known sum formula.If T denotes a tilting module either for G or Uq then we can similarly filter the space HomG(V,T), respectively HomUq(V,T) and there is a sum formula for the positive terms here as well.We give an easy and unified proof of these two (equivalent) sum formulas. Our approach is based on an Euler type identity which we show holds without any restrictions on p or l. In particular, we get rid of previous such restrictions in the tilting module case.     

On representation varieties of free abelian groups

The reducibility of the representation variety of a free abelian group of finite rank in a semisimple non-simply connected algebraic group is proved. Irreducible components of the representation variety of a free abelian group of rank 2 in groups of type A_n are described.     

Non-symmetric Cauchy kernels for the classical groups

We give non-symmetric versions of the Cauchy kernel and Littlewood's kernels, corresponding to the types A, B, C and D, of the classical groups. Defining two families of key polynomials (one of them being the Demazure characters), we show that these new kernels are diagonal in the basis of key polynomials. We define scalar products such that the two families of key polynomials are adjoint to each other.     

The normalized cyclomatic quotient associated with presentations of finitely generated groups

Given a presentation of ann-generated group, we define the normalized cyclomatic quotient (NCQ) of it, which gives a number between 1−n and 1. It is computed through an investigation of the asymptotic behavior of a kind of an “average rank”, or more precisely the quotient of the rank of the fundamental group of a finite subgraph of the corresponding Cayley graph by the size of the subgraph. In many ways (but not always) the NCQ behaves similarly to the behavior of the spectral radius of a symmetric random walk on the graph. In particular, it characterizes amenable groups. For some types of groups, like finite, amenable or free groups, its value equals that of the Euler characteristic of the group. We give bounds for the value of the NCQ for factor groups and subgroups, and formulas for its value on direct and free products. Some other asymptotic invariants are also discussed.     

Unique Node Extremal Subgroups in Chevalley Groups

Abstract

Taking G to be a Chevalley group of rank at least 3 and U to be the unipotent radical of a Borel subgroup B,an extremal subgroup A is an abelian normal subgroup of U which is not contained in the intersection of all the unipotent radicals of the rank 1 parabolic subgroups of G containing B. If there is an unique rank 1 parabolic subgroup P of G containing B with the property that A is not contained in the unipotent radical of P,then A is called a unique node extremal subgroup. In this paper we investigate the embedding of unique node extremal subgroups in U and prove that,apart from some specified cases,such a subgroup is contained in the unipotent radical of a certain maximal parabolic subgroup.