Dimension Subgroups Research Articles

Discreteness of subgroups of $PU(1,n;\mathbf{C})$

In this paper, we discuss three discreteness criterions of $n$-dimensional subgroup $G$ of $PU(1,n;\mathbf{C})$. This generalize some discreteness criterions established by J. Gilman [3], S. Yang and A. Fang [9].

CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ n+1 admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ n+1 with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group.

Classification of nonproduct states with maximum stabilizer dimension

Nonproduct n-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized n-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for n{>=}3, n{ne}4. We characterize the Lie algebra of the stabilizer subgroup for these states. For n=4, there is an additional maximal stabilizer subalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well.

Periodic Linear Groups with the Weak Chain Conditions on Subgroups of Infinite Central Dimension

Let V be a vector space over a field F. If G ≤ GL(V, F), the central dimension of G is the F-dimension of the vector space V/C V (G). In Dixon et al. (2004) and Kurdachenko and Subbotin (2006), soluble linear groups in which the set ℒ icd(G) of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. In this article we study periodic locally radical linear groups, in which the set ℒ icd(G) satisfies one of the weak chain conditions: the weak minimal condition or the weak maximal condition.

Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension

Let $V$ be a vector space over a field $F$. If $G \leq GL(V,F)$, \emph{the central dimension of $G$} is the $F$-dimension of the vector space $V/C_V(G)$. In [DEK] and [KS], soluble linear groups in which the set $\mathcal{L}_{\operatorname{icd}}(G)$ of all proper infinite central dimensional subgroups of $G$ satisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally radical linear groups in which $\mathcal{L}_{\operatorname{icd}}(G)$ satisfies one of the weak chain conditions (the weak minimal condition or the weak maximal condition) have been characterized. In this paper, we begin the study of the non-periodic case by describing locally nilpotent linear groups in which $\mathcal{L}_{\operatorname{icd}}(G)$ satisfies one of the two weak chain conditions.

Die böse Farbe

ZusammenfassungWir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. Dies beantwortet eine alte Frage von Zilber.

Finite-type invariants of 3-manifolds and the dimension subgroup problem

For a certain class of compact oriented 3-manifolds, Goussarov and Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ‘cut-and-paste’ operations defined by the lower central series of the Torelli group of a surface. In this paper, we observe that this is a variation of a classical problem in group theory, namely the ‘dimension subgroup problem’. This viewpoint allows us to prove, by purely algebraic methods, an analog of the Goussarov–Habiro conjecture for finite-type invariants with values in a fixed field. We deduce that their original conjecture is true at least in a weaker form.

Linear groups with rank restrictions on the subgroups of infinite central dimension

The authors study linear groups of infinite central dimension and of infinite p -rank all of whose proper subgroups of infinite p -rank are of finite central dimension.

The quasi-variety of groups with trivial fourth dimension subgroup

Abstract We prove that the quasi-variety consisting of the groups with trivial fourth dimension subgroup is not finitely based. This answers a problem of B. I. Plotkin.

A discrete criterion in $\mathit{PU}(2,1)$ by use of elliptic elements

In this paper we show a $2$-dimensional subgroup in $\mathit{PU}(2,1)$ which contains elliptics is discrete if and only if all its subgroups generated by two elliptics are discrete. This generalize the well-known discreteness criterion first established by T. Jørgensen.

Linear groups with the maximal condition on subgroups of infinite central dimension

Let A a vector opace over a ficid E and let H be a subgroop of GL(F, A). We define centdimF H tu be dimF(A/CA(H)). Wc say tbat H bas finite central dimeosion if centdimF H is finite and we say tbat H bao inJlnite central dimeosiso otherwise. We consider soluble linear groups, in which the (ordered by inclusion) set of ah subgroups baving infinite central dimension satisfies tbe maximal condition.

Linear groups with minimality condition for some infinite-dimensional subgroups

Let F be a field, let A be a vector space over F, and let GL(F, A) be the group of all automorphisms of the space A. If H is a subgroup of GL(F, A), then we set aug dimF (H) = dimF (A(ωFH)), where ωFH is the augmentation ideal of the group ring FH. The number aug dimF (H) is called the augmentation dimension of the subgroup H. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension.

The Cohomology of Pro-p-Groups with Group Ring Coefficients and Virtual Poincare Duality

The relationship between the group-theoretic properties of a pro-p-group G and the G-module structure of the group $$H^n (G,\mathbb{F}_q \left[\kern-0.15em\left[ G \right]\kern-0.15em\right])$$ is studied. A necessary and sufficient condition for a pro-p-group G to contain an open Poincare subgroup of dimension n is obtained. This condition does not require that G have finite cohomological dimension and, therefore, applies to groups with torsion. Results concerning the possible values of $$\dim _{\mathbb{F}p} H^n (G,\mathbb{F}_p \left[\kern-0.15em\left[ G \right]\kern-0.15em\right])$$ are also obtained.

LESS IS MORE: NONRENORMALIZATION THEOREMS FROM LOWER DIMENSIONAL SUPERSPACE

We discuss a new class of non-renormalization theorems in [Formula: see text] and [Formula: see text] Super-Yang-Mills theory, obtained by using a superspace which makes a lower dimensional subgroup of the full supersymmetry manifest. Certain Wilson loops (and Wilson lines) belong to the chiral ring of the lower dimensional supersymmetry algebra, and their expectation values can be computed exactly.

Jennings' theorem for p-split groups

Jennings [Trans. Amer. Math. Soc. 50 (1941) 175–185] used the descending central series of dimension subgroups to produce a basis of the radical layers of the group ring of a p-group over a field k of characteristic p and, with it, a Hilbert polynomial for the dimensions of these layers. Later, Quillen [J. Algebra 10 (1968) 411–418] employed Lie ring methods of Lazard [Ann. Sci. Ecole Norm. Sup. (3) 71 (1954) 101–190] to refine the proof of Jennings' theorem. Analogously, Alperin [Quart. J. Math. Oxford Ser. (2) 39 (154) (1988) 129–133] extended Jennings' methods to permutation modules for p-groups, while Shalev [Proc. Amer. Math. Soc. 120 (2) (1994) 333–337] later recast this in terms of Quillen's formulation. All of this naturally extends to p-split groups, finite groups with normal p-Sylow. Consider the filtration of the group ring kG by the powers of its Jacobson radical. When G is a p-group, Quillen showed that the associated graded ring is isomorphic to the graded p-restricted universal enveloping algebra UJ ∗ ⊗ k ( G ) for the graded p-restricted Lie algebra generated from the filtration of G by its descending central series of dimension subgroups. For a p-split group G with p-Sylow P and a given maximal subgroup A of order prime to p, the associated graded ring is isomorphic to the skew group ring UJ ∗ ⊗ k ( P ) ⋊ A (Theorem 4), where A is concentrated in degree zero and the action of A on UJ ∗ ⊗ k ( P ) is induced from its conjugation action on P. Let H be a subgroup of G and S a semisimple kH-module. Consider the graded module associated to the filtration by the powers of the Jacobson radical of the module formed by inducing S up to PH. This graded module will be isomorphic to the Hopf algebra tensor product of S (regarded as a semisimple k P H -module) with the quotient of UJ ∗ ⊗ k ( P ) by a left ideal generated by elements corresponding to q − 1 , where the q are elements of the p-Sylow of H; inducing this graded module up to G will preserve the grading (Theorem 5). Hilbert polynomials for the Brauer character of the radical layers allow explicit computation of the isomorphism classes of the radical layers of the module produced by inducing S to G.

Dimension and randomness in groups acting on rooted trees

We explore the structure of thepp-adic automorphism groupYYof the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups ofWWgenerated by three random elements are full dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.

Linear groups with the minimal condition on subgroups of infinite central dimension

The authors study infinite dimensional linear groups with the minimal condition on subgroups of infinite central dimension.

Linear groups with the minimal condition on subgroups of infinite central dimension

The authors study infinite dimensional linear groups with the minimal condition on subgroups of infinite central dimension.

Augmentation powers and group homology

Generalizing the notion of polynomial 2-cocycles, a homological approach to the identification of normal subgroups determined by two-sided ideals in group rings is developed. In particular, transfinite dimension subgroups, i.e., the subgroups determined by the transfinite augmentation powers are investigated.

The maximal subgroups of positive dimension in exceptional algebraic groups

Introduction Preliminaries Maximal subgroups of type $A_1$ Maximal subgroups of type $A_2$ Maximal subgroups of type $B_2$ Maximal subgroups of type $G_2$ Maximal subgroups $X$ with rank$(X)\geq3$ Proofs of Corollaries 2 and 3 Restrictions of small $G$-modules to maximal subgroups The tables for Theorem 1 and Corollary 2 Appendix: $E_8$ structure constants References.