Congruence Kernels Research Articles

On the Congruence Subgroup Problem for integral group rings

Let G be a finite group, ZG the integral group ring of G and U(ZG) the group of units of ZG. The Congruence Subgroup Problem for U(ZG) is the problem of deciding if every subgroup of finite index of U(ZG) contains a congruence subgroup, i.e. the kernel of the natural homomorphism U(ZG)→U(ZG/mZG) for some positive integer m. The congruence kernel of U(ZG) is the kernel of the natural map from the completion of U(ZG) with respect to the profinite topology to the completion with respect to the topology defined by the congruence subgroups. The Congruence Subgroup Problem has a positive solution if and only if the congruence kernel is trivial. We obtain an approximation to the problem of classifying the finite groups for which the congruence kernel of U(ZG) is finite. More precisely, we obtain a list L formed by three families of finite groups and 19 additional groups such that if the congruence kernel of U(ZG) is infinite then G has an epimorphic image isomorphic to one of the groups of L. Regarding the converse of this statement we at least know that if one of the 19 additional groups in L is isomorphic to an epimorphic image of G then the congruence kernel of U(ZG) is infinite. However, to decide for the finiteness of the congruence kernel in case G has an epimorphic image isomorphic to one of the groups in the three families of L one needs to know if the congruence kernel of the group of units of an order in some specific division algebras is finite and this seems a difficult problem.

Normal subgroups of profinite groups of non-negative deficiency

The principal goal of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group G of non-negative deficiency gives rather strong consequences for the structure of G. To make this precise we introduce the notion of p-deficiency (p a prime) for a profinite group G. We prove that if the p-deficiency of G is positive and N is a finitely generated normal subgroup such that the p-Sylow subgroup of G/N is infinite and p divides the order of N then we have cdp(G)=2, cdp(N)=1 and vcdp(G/N)=1 for the cohomological p-dimensions; moreover either the p-Sylow subgroup of G/N is virtually cyclic or the p-Sylow subgroup of N is cyclic. If G is a profinite Poincaré duality group of dimension 3 at a prime p (PD3-group at p) we show that for N and p as above either N is PD1 at p and G/N is virtually PD2 at p or N is PD2 at p and G/N is virtually PD1 at p.We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups and 3-manifold groups. We prove that the arithmetic lattices in SL2(C) are cohomologically good and give some implications of our theory to their congruence kernels.

Ideals and congruences of basic algebras

MV-algebras as well as orthomodular lattices can be seen as a particular case of so-called "basic algebras" which are an alter ego of bounded lattices whose sections are equipped with fixed antitone involutions. The class of basic algebras is an ideal variety. In the paper, we give an internal characterization of congruence kernels (ideals) and find a finite basis of ideal terms, with focus on monotone and effect basic algebras. We also axiomatize basic algebras that are subdirect products of linearly ordered ones.

The congruence subgroup problem for branch groups

We state and study the congruence subgroup problem for groups acting on rooted trees, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels. We work out explicitly known and new examples of non-trivial congruence kernels, describing in each case the group action. The Hanoi tower group receives particular attention due to its surprisingly rich behaviour.

Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank >1 over noetherian rings

Let G G be a universal Chevalley-Demazure group scheme associated to a reduced irreducible root system of rank > 1. >1. For a commutative ring R R , we let Γ = E ( R ) \Gamma = E(R) denote the elementary subgroup of the group of R R -points G ( R ) . G(R). The congruence kernel C ( Γ ) C(\Gamma ) is then defined to be the kernel of the natural homomorphism Γ ^ → Γ ¯ , \widehat {\Gamma } \to \overline {\Gamma }, where Γ ^ \widehat {\Gamma } is the profinite completion of Γ \Gamma and Γ ¯ \overline {\Gamma } is the congruence completion defined by ideals of finite index. The purpose of this paper is to show that for an arbitrary noetherian ring R R (with some minor restrictions if G G is of type C n C_n or G 2 G_2 ), the congruence kernel C ( Γ ) C(\Gamma ) is central in Γ ^ . \widehat {\Gamma }.

Congruence kernels of distributive PJP-semilattices

A meet semilattice with a partial join operation satisfying certain axioms is a JP-semilattice. A PJP-semilattice is a pseudocomplemented JP-semilattice. In this paper we describe the smallest PJP-congruence containing a kernel ideal as a class. Also we describe the largest PJP-congruence containing a filter as a class. Then we give several characterizations of congruence kernels and cokernels for distributive PJP-semilattices.

A non commutative generalization of *-autonomous lattices

Pseudo *-autonomous lattices are non-commutative generalizations of *-autonomous lattices. It is proved that the class of pseudo *-autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo *-autonomous lattices can be described as their normal ideals.

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let kυ be the completion of k with respect to υ a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of kυ-rank 1. Let G = G(kυ). Let Г be an arithmetic lattice in G and let C = C(Г) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Г by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is , the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example , where is the ring of S-integers in k, with S = {υ}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Г on the Bruhat-Tits tree associated with G.

Congruence kernels of orthomodular implication algebras

Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthomodular implication algebras. These are in a natural one-to-one correspondence with families of orthomodular lattices. It is proved that congruence kernels of orthomodular implication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices.

Distributive implication groupoids

We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.

Subdirectly irreducible sectionally pseudocomplemented semilattices

Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented semilattices.

Profinite surface groups and the congruence kernel of arithmetic lattices in SL2 (R)

LetX be a proper, nonsingular, connected algebraic curve of genusg over the fieldC of complex numbers. The algebraic fundamental group Γ = π1 (X) in the sense of [SGA-1] (1971) is the profinite completion of the fundamental group π1top (X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, characteristic, accessible) subgroup of Γ is isomorphic to a normal (respectively, characteristic, accessible) subgroup of a free profinite group. We use this description to give a complete solution of the congruence subgroup problem for arithmetic lattices in SL2 (R).

An Implication in Orthologic

We involve a certain propositional logic based on an ortholattice. We characterize the implication reduct of such a logic and show that its algebraic counterpart is the so-called orthosemilattice. Properties of congruences and congruence kernels of these algebras are described.

Congruences, ideals and annihilators in standard QBCC-algebras

Abstract We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels.

Finite Presentations of adelic groups, the congruence kernel and cohomology of finite simple groups

Let K be a non-archimedean local field and U a compact subgroup of GLn(K). Such U is a profinite group, actually it is virtually pro-p. If char(K) = 0, then U is finitely generated and even finitely presented as a profinite group. On the other hand if char(K) = p > 0, then U need not be finitely generated. Moreover, even if U is finitely generated, it may be non-finitely presented (see(1.3)). On the other hand;

Annihilators on weakly standard BCC‐algebras

In a recent paper the authors presented a new construction of BCC‐algebras derived from posets with the top element 1. Resulting BCC‐algebras, called weakly standard, are those for which every 4‐element subset containing 1 is a subalgebra. In this paper we continue our investigations focusing on the properties of their lattices of congruence kernels.

Pseudo $BL$-algebras and $DR\ell $-monoids

It is shown that pseudo $BL$-algebras are categorically equivalent to certain bounded $DR\ell $-monoids. Using this result, we obtain some properties of pseudo $BL$-algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo $BL$-algebras and, in conclusion, we prove that they form a variety.

Maps between Jacobians of Shimura curves and congruence kernels

Let p ≥ 5 be a prime, and℘ a place ofQ above it.We denote byGQ the absolute Galois group Gal(Q/Q). For a finite field k of characteristic p, an absolutely irreducible representation ρ : GQ → GL2(k) is said to arise from a newform of weight k and level a positive integer M if ρ is isomorphic to the reduction mod ℘ of the ℘-adic representation associated to a newform of weight k and level M . Note that then it is known that ρ also arises from a newform of weight 2 and level dividingMp2. Thus we assume without loss of generality that ρ arises from a newform of weight 2 and level a positive integer N . The question of “raising” the level of ρ, i.e., determining when a ρ arising from a newform of weight 2 and level N as above also arises from a newform of weight 2 and levelN ′, whereN dividesN ′, has been widely studied by numerous mathematicians (cf. [10,8,5,2]). In the first paper that considers this kind of question [10], as well as in most of the papers dealing with similar questions (e.g., [2]), the cases considered correspond to N ′/N being coprime to p. Some discussion for the case where

Congruence Kernels in Weakly Regular Varieties

Congruence Kernels in Weakly Regular Varieties

Congruence Kernels in Weakly Regular Varieties

We generalize the concept of deductive system of Hilbert algebras for the general case of algebras in weakly regular variety and we show that congruence kernels of these algebras are just deductive systems. This concept enables us to describe congruence classes in algebras of weakly regular varieties.