Finite Quotient Group Research Articles

Cocycle superrigidity for profinite actions of irreducible lattices

Let \Gamma be an irreducible lattice in a product of two locally compact groups and assume that \Gamma is densely embedded in a profinite group K . We give necessary conditions which imply that the left translation action \Gamma \curvearrowright K is “virtually” cocycle superrigid: any cocycle {w\colon \Gamma\times K\rightarrow\Delta} with values in a countable group \Delta is cohomologous to a cocycle which factors through the map \Gamma\times K\rightarrow\Gamma\times K_0 for some finite quotient group K_0 of K . As a corollary, we deduce that any ergodic profinite action of \Gamma=\mathrm{SL}_2(\mathbb Z[S^{-1}]) is virtually cocycle superrigid and virtually W ^* -superrigid for any finite nonempty set of primes S .

Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups

For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.

Finite quotients of groups of I-type

To every group of I-type, we associate a finite quotient group that plays the role that Coxeter groups play for Artin–Tits groups. Since groups of I-type are examples of Garside groups, this answers a question of D. Bessis in the particular case of groups of I-type. Groups of I-type are related to finite set-theoretical solutions of the Yang–Baxter equation. So, our result provides a new tool to attack the problem of the classification of finite set-theoretical solutions of the Yang–Baxter equation.

The intersection condition for regular polytopes

The automorphism group of every abstract regular polytope of type {k1|⋯|km} is a smooth finite quotient of the corresponding Coxeter group [k1,…,km] satisfying a particular condition on subgroups known as the intersection condition (or intersection property), and vice versa. In this paper, we consider the question of how many cases of the intersection condition need to be checked, sometimes under additional hypotheses about the finite quotient group or the type of the polytope. We also apply this to the construction of a family of polytopal regular maps of every possible genus, and an infinite family of locally spherical regular 4-polytopes of type {3|5|3}, with all but finitely many alternating groups as automorphism groups.

Periods of generic torsors of groups of multiplicative type

In this paper we establish different ways to compute the period of a generic torsor of a group of multiplicative type. We also study the relationships among the periods of generic torsors of Q, T and W, where Q is a group of multiplicative type, T is the maximal subtorus of Q and W is the maximal finite quotient group of Q.

Approximation on the Voronoi Cells of the A d Lattice

We study several approximation problems on the Voronoi cells of the Ad lattice. The study depends on discretization of the Voronoi cells induced by a zonotopal algebra and finite quotient groups. The latter provides a natural tool for doing discrete Fourier analysis on the Voronoi cells. The interactions between the groups and their dual groups yield interpolation and quadrature formulas on the Voronoi cells. The zonotope structure allows us to investigate the approximation property of several types of kernels (including the Dirichlet type and the Fejer type) on the Voronoi cells. The convolution operator given by the Fejer type kernel is positive, and consequently an approximate identity. The Fejer type kernel can also be realized as a summability method. As an interesting comparison to the classical (C,1) summability which assigns equal weight to the partial sums of a Fourier series, the Fejer type summability method on the Voronoi cells assigns equal weights to about half of the partial sums and algebraically decaying weights to the rest.

Linear Representations of the Automorphism Group of a Free Group

Let Fn be the free group on n ≥ 2 elements and Aut(Fn) its group of automorphisms. In this paper we present a rich collection of linear representations of Aut(Fn) arising through the action of finite-index subgroups of it on relation modules of finite quotient groups of Fn. We show (under certain conditions) that the images of our representations are arithmetic groups.

Equivariant homotopy theory for pro–spectra

We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G-homotopy theory is "pieced together" from the G/U-homotopy theories for suitable quotient groups G/U of G; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro-G-spectra and construct various model structures on them. A key property of the model structures is that pro-spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with "underlying weak equivalences". One of the versions only makes sense for pro-spectra. In the end we use the theory to study homotopy fixed points of pro-G-spectra.

Some algebraic properties of crystallographic sublattices

In this article, a number of the results relevant to the concept of sublattices of a basic crystallographic lattice are reviewed, emphasizing particularly previously unpublished work on the algebraic aspects. A three-dimensional geometric lattice L can be considered as an infinite Abelian group under addition. A sublattice S of L, which is also three-dimensional, is a subgroup of L such that the finite quotient group, G approximately equals L/S, is an Abelian group of order the index of S in L. The sublattice itself in its standard form is represented by an upper triangular matrix. The index of the sublattice is given by the determinant of this matrix. It is first noted that a sublattice described by an arbitrary basis set in L may be converted to this standard form. Next the sublattice is expressed as the intersection of a set of sublattices of individual index a power of a distinct prime, i.e. S(n = p(a)(1)p(b)(2)...) = S(1)(p(a)(1)[cap]S(2)(p(b)(2)...[cap]... = [bigcap](i)S(i)(p(alpha(i)), where p(1), p(2) etc. are prime numbers and n = Pi(i)p(alpha)(i) is the Euclidean factorization of n. This decomposition is important because it corresponds to the Sylow decomposition of the corresponding quotient group G approximately equals (i)[sign: see text] A(p)(i). It is also useful to be able to carry out two commutative binary operations on sublattices of L; these are to find their common sublattice of lowest index in L, which is their intersection S(cap) = S(a)(m)[cap]S(b)(n) and their common superlattice of highest index in L, given by S(< >) = <S(a)(m), S(b)(n)>, where < > indicates the span of the sublattices.

How often is 84(g−1) achieved?

For any finitely generated group Γ, the asymptotics of the set of orders of finite quotient groups of Γ are determined by the minimum dimension of a complex linear group containing an infinite quotient of Γ. We give a proof and an application to the asymptotic behavior of the set of integersg for which the Hurwitz bound is sharp.

Groups all proper quotient groups of which have Chernikov conjugacy classes

Numerous problems in group theory require the investigation of the influence of certain systems of proper quotient groups (quotient groups modulo nonidentity normal subgroups) on the structure of the entire group. Thus, the family of all finite quotient groups plays an important role in the investigation of algorithmic problems in finitely defined groups (Maltsev [1]) and in the theory of manifolds [2]. The important Robinson theorem on the nilpotency of a finitely generated soluble group all finite quotient groups of which are nilpotent (see Theorem 10.51 in [3]) also demonstrates the role of the family of finite quotient groups. Quotient groups modulo centralizers of principal quotients also play an important role in the theory of formations. On the other hand, the investigation of the structure of a group with the given family of all proper quotient groups has a comparatively long history. For the first time, problems of this type in the theory of infinite groups were considered by Newman in [4, 5], where the groups with Abelian proper quotient groups were studied. Later, investigations in this direction were continued for various group-theoretic properties: McCarthy [6, 7] and Wilson [8] studied groups all proper quotient groups of which are finite, Franciosi and de Giovanni [9] and Kurdachenko, Goretskii, and Pylaev [10] studied groups with Chernikov proper quotient groups, Groves [11] and Robinson and Wilson [ 12] investigated groups with proper polycyclic quotient groups, and Robinson and Zhang [ 13] considered groups all proper quotient groups of which are finite over the centers or have finite commutants. The investigations of Robinson and Zhang were continued by Franciosi, de Giovanni, and Kurdachenko in [14], where the groups all proper quotient groups of which are FC-groups were considered. At present, there exist various generalizations of FC-groups defined as follows: Let Y be a class of groups. We say that G is a ,~C-group (or that G has con-

Counting subgroups of non-euclidean crystallographic groups

A method is obtained for counting the normal subgroups N of a noneuclidean crystallographic groupwithout reflections, with a given finite quotient group ˇ=N; this has applications to the enumeration of regular coverings of orbifolds. The method, which involves Mo« bius inversion and character theory, is also applied to count normal surface subgroups and non-normal sub- groups of finite index in ˇ.

On the Conjugacy Separability of Certain Graphs of Groups

A group G is said to be conjugacy separable if, whenever x and y are nonconjugate elements of G, there exists a finite quotient group of G in which the images of x and y are not conjugate. The importance of this w x notion was pointed out by Mal’cev, who proved in 7 that if a finitely presented group G is conjugacy separable, then G has solvable the conjugacy problem, that is, there exists an algorithm to decide whether or not any two given elements of G are conjugate. Žw x. Žw x. Polycyclic-by-finite groups 5, 12 , free groups 13, 16 , free-by-finite w x groups 3 are conjugacy separable. It is known that free products of Žw x. conjugacy separable groups are again conjugacy separable 16, 13 . However, the property is not preserved in general by the formation of free products with amalgamation or HNN-extensions. For example, one of the simplest type of HNN-extensions, the Baum2 y1 7 13: slag]Solitar group a, t ¬ t a t s a is not even residually finite; note that if a group G is conjugacy separable then it must be residually finite. w x C. F. Miller in 8 gave examples of HNN-extensions which are residually finite and not conjugacy separable. In this paper we are concerned with the conjugacy separability of fundamental groups of certain graphs of groups. Our main results are: