Finite Cyclic Subgroups Research Articles

Random walks on the circle and Diophantine approximation.

Random walks on the circle group whose elementary steps are lattice variables with span or taken mod exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup , and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper, we extend these results to random walks with irrational span , and explicitly describe the transition of these Markov chains from finite to general state space as along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a phase transition from polynomial to exponential decay after steps. This seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purelyexponential.

Pursuing quantum difference equations I: stable envelopes of subvarieties

Let X be a symplectic variety equipped with an action of a torus \({\mathsf {A}}\). Let \({\varvec{\nu }}_{b}\subset {\mathsf {A}}\) be a finite cyclic subgroup. We show that K-theoretic stable envelope of the fixed point set \(X^{{\varvec{\nu }}_{b}}\subset X\) can be obtained via a limit of the elliptic stable envelopes of X. An example of X given by the Hilbert scheme of points in the complex plane is considered in detail.

On Finite Subgroups in the General Linear Groups over an Algebraic Number Field

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

English

Let α and β be automorphisms of a nilpotent p-group G of finite rank. Suppose that 〈(αβ(g))(βα(g))–1: g ∈G〉 is a finite cyclic subgroup of G, then, exclusively, one of the following statements holds for G and Γ, where Γ is the group generated by α and β.

Ultrasoluble coverings of some nilpotent groups by a cyclic group over number fields and related questions

Let be a finite nilpotent group of odd order. For every finite cyclic subgroup of odd order we find necessary and sufficient conditions for a class to determine an ultrasoluble extension (under the additional assumption of minimality of all -Sylow subextensions to the extension with class for all non-Abelian -Sylow subgroups of ), that is, for the existence of a Galois extension of number fields with group such that the corresponding embedding problem is ultrasoluble (it has solutions and all such solutions are fields). We also establish a number of related results.

О ПРОБЛЕМЕ ПЕРЕСЕЧЕНИЯ КЛАССОВ СМЕЖНОСТИ КОНЕЧНО ПОРОЖДЕННЫХ ПОДГРУПП В ГРУППЕ КОКСТЕРА С ДРЕВЕСНОЙ СТРУКТУРОЙ

P. S. Novikov in 1955–1956 showed the unsolvability of the main algorithmic problems in class of finite defined groups. In this connection there was important task of consideration of these problems in specific classes of finite defined groups. Thus, class of finite defined groups of Coxeter represents scientific interest. The class of groups of Coxeter was defined by H. S. M. Coxeter in 1934. The classe of finitely generated groups of Coxeter with tree structure was defined by V. N. Bezverkhnii in 2003. Let finitely generated group of Coxeter with tree structure is defined by presentation G = ha1, ...an; (ai) 2 ,(aiaj ) mij , i, j ∈ 1, n, i 6= ji where mij — number which corresponds to a symmetric matrix of Coxeter. At that, if i 6= j, that mij = mji, mij > 2. If mij = ∞, that between ai and aj relation does not exist . The group matches finite coherent tree-graph Γ such that: if tops of some edge -e of graf Г are elements ai and aj , that the edge e matches relation (aiaj ) mij = 1. On the other hand group G may be represented as wood product of the two-generated groups of Coxeter, which are united by final cyclic subgroups. In this case, we will pass from graf Г of group G to graf Γ as follows: we associate tops of some edge e of graf Γ groups of Coxeter with two generating elements Gji = haj , ai ; (aj ) 2 ,(ai) 2 ,(ajai) mji i and Gik = hai , ak; (ai) 2 ,(ak) 2 ,(aiak) mik i, and edge e — cyclic subgroup hai ; (ai) 2 i. The problem of intersection of the adjacency classes of finitely generated subgroups is that you need to find an algorithm that will help determine empty or not intersection w1H1 ∩ w2H2, where H1 and H2 any subgroup of group G and w1, w2 ∈ G. Previously, the author proved the solvability of this problem for free product with association of two Coxeter’s groups with two generating element. In the article author shows solvability of a problem of intersection of the adjacency classes of finite number of finitely generated subgroups of Coxeter’s group with tree structure. For this purpose group G was presented as wood product of n two-generated groups of Coxeter, which are united by finite cyclic subgroups. To prove of this result, the author used the method of special sets and method of types. These methods were defined V. N. Bezverkhnii. He used these methods for research of various algorithmic problems in free constructions of groups.

Spectrum of a composition operator with automorphic symbol

We give a complete characterization of the spectrum of composition operators, induced by an automorphism of the open unit disk, acting on a family of Banach spaces of analytic functions that includes the Bloch space and BMOA. We show that for parabolic and hyperbolic automorphisms, the spectrum is the unit circle. For the case of elliptic automorphisms, the spectrum is either the unit circle or a finite cyclic subgroup of the unit circle.

A heuristic approach for designing cyclic group codes

AbstractIn this paper, we propose a heuristic technique for distributing points on the surface of a unit n‐dimensional Euclidean sphere, generated as the orbit of a finite cyclic subgroup of orthogonal matrices, the so‐called cyclic group codes. Massive numerical experiments were conducted and many new cyclic group codes have been obtained in several dimensions at various rates. The results assure that the performance of the heuristic approach is comparable to a brute‐force search technique with the advantage of having low complexity, which allows designing codes with a large number of points in high dimensions.

The combinatorics of certain group-invariant mappings

We study combinatorial and number-theoretic issues arising from group-invariant CR mappings from spheres to hyperquadrics. Given an initial complex analytic function, we define two sequences of complex numbers. We derive formulae for these sequences in terms of the Taylor coefficients of the initial function and, in the polynomial case, also in terms of the reciprocals of the roots. We connect these formulae to CR mappings invariant under arbitrary representations of finite cyclic subgroups of the unitary group. We prove that the resultant of the m-th polynomial (which in all cases has integer coefficients) and any of its partial derivatives is divisible by m m .

McKay-type correspondence for AS-regular algebras

The classical McKay correspondence claims that the minimal resolution of Spec k[x, y]G where G ⩽ SL(2, k) is a finite subgroup naturally acting on k[x, y] is derived equivalent to the preprojective algebra of the McKay quiver of G. In this paper, we will see that there is a similar correspondence in the case of a finite cyclic subgroup G ⩽ GL(n, k) naturally acting on an AS-regular algebra in n variables. Since AS-regular algebras are non-commutative analogues of polynomial algebras, this can be thought of as a McKay-type correspondence in non-commutative algebraic geometry.

Knit Products of Some Groups and Their Applications

Let G be a group with subgroups A and K (not necessarily normal) such that G = AK and A \cap K = \{1\} . Then G is isomorphic to the knit product , that is, the “two-sided semidirect product” of K by A . We note that knit products coincide with Zappa-Szep products (see [18]). In this paper, as an application of [2, Lemma 3.16], we first define a presentation for the knit product G where A and K are finite cyclic subgroups. Then we give an example of this presentation by considering the (extended) Hecke groups. After that, by defining the Schur multiplier of G , we present sufficient conditions for the presentation of G to be efficient. In the final part of this paper, by examining the knit product of a free group of rank n by an infinite cyclic group, we give necessary and sufficient conditions for this special knit product to be subgroup separable.

Existence and Non-Existence of Torsion in Maximal Arithmetic Fuchsian Groups

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

A contact-stationary Legendrian submanifold of \({\mathbb {S}^{2n+1}}\) is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S0. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of \({\mathbb {S}^{2n+1}}\) by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S0. Two examples of the construction, corresponding to finite cyclic subgroups of U(n + 1) are given. The resulting submanifolds are very symmetric; are geometrically akin to a ‘necklace’ of copies of S0 attached to each other by narrow necks and winding a large number of times around \({\mathbb {S}^{2n+1}}\) before closing up on themselves; and are topologically equivalent to \({\mathbb {S}^1 \times \mathbb {S}^{n-1}}\) .

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.

Recollement of Deformed Preprojective Algebras and the Calogero–Moser Correspondence

The aim of this paper is to clarify the relation between the following objects: (a) rank 1 projective modules (ideals) over the first Weyl algebra A_1; (b) simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and (c) simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on A-infinity modules over A_1 to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $C^2/\Gamma$, where $\Gamma$ is a finite cyclic subgroup of $SL(2,C)$.

Orbifold index and equivariant K-homology

Let G be countable group and M be a proper cocompact even-dimensional G-manifold with orbifold quotient \(\bar{M}\) . Let D be a G-invariant Dirac operator on M. It induces an equivariant K-homology class \([D] \in K^G_0(M)\) and an orbifold Dirac operator \(\bar{D}\) on \(\bar{M}\) . Composing the assembly map \(K^G_0(M) \rightarrow K_0(C*(G))\) with the homomorphism \(K_0(C^*(G)) \rightarrow {\mathbb{Z}}\) given by the representation \(C^*(G) \rightarrow {\mathbb{C}}\) of the maximal group C *-algebra induced from the trivial representation of G we define index\(([D]) \in {\mathbb{Z}}\) . In the second section of the paper we show that index\((\bar{D})\) = index([D]) and obtain explicit formulas for this integer. In the third section we review the decomposition of \(K_0^G(M)\) in terms of the contributions of fixed point sets of finite cyclic subgroups of G obtained by W. Lück. In particular, the class [D] decomposes in this way. In the last section we derive an explicit formula for the contribution to [D] associated to a finite cyclic subgroup of G.

The behavior of Nil-groups under localization and the relative assembly map

We study the behavior of the Nil-subgroups of K -groups under localization. As a consequence of our results, we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character, we obtain a complete computation of the rationalized source of the K -theoretic assembly map that appears in the Farrell–Jones conjecture in terms of group homology and the K -groups of finite cyclic subgroups. Specifically we prove that under mild assumptions we can always write the Nil-groups and End-groups of the localized ring as a certain colimit over the Nil-groups and End-groups of the ring, generalizing a result of Vorst. We define Frobenius and Verschiebung operations on certain Nil-groups. These operations provide the tool to prove that Nil-groups are modules over the ring of Witt-vectors and are either trivial or not finitely generated as Abelian groups. Combining the localization results with the Witt-vector module structure, we obtain that Nil and localization at an appropriate multiplicatively closed set S commute, i.e. S − 1 Nil = Nil S − 1 . An important corollary is that the Nil-groups appearing in the decomposition of the K -groups of virtually cyclic groups are torsion groups.

Rational computations of the topological K-theory of classifying spaces of discrete groups

We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.

Induction Theorems and Isomorphism Conjectures for K- and L-Theory

The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C -algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with Z as kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.

Equivariant Euler characteristics andK–homology Euler classes for proper cocompactG–manifolds

Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant K-homology of M. The universal equivariant Euler characteristic of M, which lives in a group U^G(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from U^G(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no `higher' equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the L-fixed point sets M^L, where L runs through the finite cyclic subgroups of G. However, we give an example of an action of the symmetric group S_3 on the 3-sphere for which the equivariant Euler class has order 2, so there is also some torsion information.