Congruence Subgroup Problem Research Articles

Multiplier systems for Siegel modular groups

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

The congruence subgroup problem for finitely generated nilpotent groups

Abstract The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) G\leq\mathrm{Aut}(\Gamma) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) \hat{G}\to\mathrm{Aut}(\hat{\Gamma}) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) C(G,\Gamma) is. Here X ^ \hat{X} denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) G=\mathrm{Aut}(\Gamma) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) C(\Gamma)=C(\mathrm{Aut}(\Gamma),\Gamma) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] \bar{\Gamma}=\Gamma/[\Gamma,\Gamma] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) \Gamma^{*}=\bar{\Gamma}/\mathrm{tor}(\bar{\Gamma})\cong\mathbb{Z}^{(d)} . Define Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) . \mathrm{Aut}^{*}(\Gamma)=\operatorname{Im}(\mathrm{Aut}(\Gamma)\to\mathrm{Aut}(\Gamma^{*}))\leq\mathrm{GL}_{d}(\mathbb{Z}). In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) . C(\Gamma)\simeq C(\mathrm{Aut}^{*}(\Gamma),\Gamma^{*}). In other words, C ⁢ ( Γ ) C(\Gamma) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut * ⁢ ( Γ ) \mathrm{Aut}^{*}(\Gamma) . In particular, in the case where Γ = Ψ n , c \Gamma=\Psi_{n,c} is a finitely generated free nilpotent group of class 𝑐 on 𝑛 elements, we get that C ⁢ ( Ψ n , c ) = C ⁢ ( Z ( n ) ) = { e } C(\Psi_{n,c})=C(\mathbb{Z}^{(n)})=\{e\} whenever n ≥ 3 n\geq 3 , and C ⁢ ( Ψ 2 , c ) = C ⁢ ( Z ( 2 ) ) = F ^ ω C(\Psi_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_{\omega} is the free profinite group on countable number of generators.

The congruence subgroup problem for the free metabelian group on $n\geq4$ generators

The congruence subgroup problem for a finitely generated group \Gamma asks whether the map \widehat{\mathrm{Aut}(\Gamma)}\to \mathrm{Aut}(\widehat{\Gamma}) is injective, or more generally, what is its kernel C(\Gamma) ? Here \widehat{X} denotes the profinite completion of X . It is well known that for finitely generated free abelian groups C(\mathbb{Z}^{n})=\{ 1\} for every n\geq3 , but C(\mathbb{Z}^{2})=\widehat{F}_{\omega} , where \widehat{F}_{\omega} is the free profinite group on countably many generators. Considering \Phi_{n} , the free metabelian group on n generators, it was also proven that C(\Phi_{2})=\widehat{F}_{\omega} and C(\Phi_{3})\supseteq\widehat{F}_{\omega} . In this paper we prove that C(\Phi_{n}) for n\geq4 is abelian. So, while the dichotomy in the abelian case is between n=2 and n\geq3 , in the metabelian case it is between n=2,3 and n\geq4 .

The IA-congruence kernel of high rank free metabelian groups

The congruence subgroup problem for a finitely generated group $\Gamma$ and $G\leq Aut(\Gamma)$ asks whether the map $\hat{G}\to Aut(\hat{\Gamma})$ is injective, or more generally, what is its kernel $C\left(G,\Gamma\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA(\Phi_{n}),\Phi_{n}\right)$, where $\Phi_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(\Phi_{n})=\ker(Aut(\Phi_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(\Phi_{n}),\Phi_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happens for free metabelian group on $n=2,3$ generators, or for finitely generated nilpotent groups.

The congruence subgroup problem for a family of branch groups

We construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups in this generalization are just infinite and have a trivial rigid kernel. We also put strict bounds on the branch kernel. Additionally, we show that these groups have subgroups of finite index with non-trivial rigid kernel. The only previously known group where this kernel is non-trivial is the Hanoi towers group and so this adds infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1.

Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro- $$\mathcal {C}$$ completions of the group, where $$\mathcal {C}$$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $$\mathcal {C}$$ -congruence subgroup property ( $$\mathcal {C}$$ -CSP) if its pro- $$\mathcal {C}$$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $$\mathcal {C}$$ -CSP. In the case where $$\mathcal {C}$$ is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.

A pro-l version of the congruence subgroup problem for mapping class groups of genus one

Let l be a prime number. In the present paper, we discuss a pro-l version of the congruence subgroup problem for mapping class groups of genus one. Our main result is that the pro-2 version has an affirmative answer, but the pro-l version for l≥11 has a negative answer.

Non-residually finite extensions of arithmetic groups

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose G is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of G has finite extensions which are not residually finite. More precisely, we investigate the group H¯2(Z/n)=limΓ→H2(Γ,Z/n),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2(\\mathbb {Z}/n) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle \\varGamma }\\limits ^{\\textstyle \\rightarrow }} \\end{array} H^2(\\varGamma ,\\mathbb {Z}/n), \\end{aligned}$$\\end{document}where varGamma runs through the arithmetic subgroups of G. Elements of {bar{H}}^2(mathbb {Z}/n) correspond to (equivalence classes of) central extensions of arithmetic groups by mathbb {Z}/n; non-zero elements of {bar{H}}^2(mathbb {Z}/n) correspond to extensions which are not residually finite. We prove that {bar{H}}^2(mathbb {Z}/n) contains infinitely many elements of order n, some of which are invariant for the action of the arithmetic completion {widehat{G(mathbb {Q})}} of G(mathbb {Q}). We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group H¯2(Zl)=limt←H¯2(Z/lt).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2({\\mathbb {Z}_l}) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle t}\\limits ^{\\textstyle \\leftarrow }} \\end{array} {\\bar{H}}^2(\\mathbb {Z}/l^t). \\end{aligned}$$\\end{document}We show that {bar{H}}^2({mathbb {Z}_l})^{widehat{G(mathbb {Q})}} is isomorphic to {mathbb {Z}_l}^c for some positive integer c. When G(mathbb {R}) has no simple components of complex type, we prove that c=b+m, where b is the number of simple components of G(mathbb {R}) and m is the dimension of the centre of a maximal compact subgroup of G(mathbb {R}). In all other cases, we prove upper and lower bounds on c; our lower bound (which we believe is the correct number) is b+m.

The congruence subgroup problem for low rank free and free metabelian groups

The congruence subgroup problem for a finitely generated group Γ asks whether Aut(Γ)ˆ→Aut(Γˆ) is injective, or more generally, what is its kernel C(Γ)? Here Xˆ denotes the profinite completion of X.In this paper we first give two new short proofs of two known results (for Γ=F2 and Φ2) and a new result for Γ=Φ3:(1)C(F2)={e} when F2 is the free group on two generators.(2)C(Φ2)=Fˆω when Φn is the free metabelian group on n generators, and Fˆω is the free profinite group on ℵ0 generators.(3)C(Φ3) contains Fˆω. Results (2) and (3) should be contrasted with an upcoming result of the first author showing that C(Φn) is abelian for n≥4.

On the congruence subgroup problem for branch groups

We answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup problem for branch groups is independent of the branch action on a tree. We prove that the congruence topology of a branch group is determined by the group; specifically, by its structure graph, an object first introduced by Wilson. We also give a more natural definition of this graph.

The congruence subgroup problem for the free metabelian group on two generators

In this paper we describe the profinite completion of the free solvable group on m generators of solvability length r \geq 1 . Then, we show that for m = r = 2 , the free metabelian group on two generators does not have the Congruence Subgroup Property.

Presentations for Quaternionic S-Unit Groups

We give an algorithm for presenting S-unit groups of an order in a definite rational quaternion algebra B such that for every p ∈ S at which B splits, the localization of at p is maximal, and all left ideals of of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamilton’s quaternions over the rational field . To our knowledge, this provides the first explicit presentations of an S-arithmetic lattice in a semisimple Lie group with S large. We also include some discussion and experimentation related to the congruence subgroup problem, which is open for S-units of the Hurwitz order when S contains at least two odd primes.

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.

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension. Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O,o), where O is an unramified quadratic extension of o. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.

Galois coverings of moduli spaces of curves and loci of curves with symmetry

Let $$\overline{\mathcal{M }}_{g,[n]}$$ , for $$2g-2+n>0$$ , be the stack of genus $$g$$ , stable algebraic curves over $$\mathbb{C }$$ , endowed with $$n$$ unordered marked points. In Looijenga (J Algebraic Geom 3:283–293, 1994), Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack $$\overline{\mathcal{M }}_{g}$$ . In Sect. 2 of this paper, we introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of $$\overline{\mathcal{M }}_{g,[n]}$$ equivalent to the tower of all geometric level structures over $$\overline{\mathcal{M }}_{g,[n]}$$ . In Sect. 3, Looijenga level structures are interpreted geometrically in terms of moduli of curves with symmetry. A byproduct of this characterization is a simple criterion for their smoothness. As a consequence of this criterion, it is shown that Looijenga level structures are smooth under mild hypotheses. The second part of the paper, from Sect. 4, deals with the problem of describing the D–M boundary of Looijenga level structures. In Sect. 6, a description is given of the nerve of the D–M boundary of abelian level structures. In Sect. 7, it is shown how this construction can be used to “approximate” the nerve of Looijenga level structures. These results are then applied to elaborate a new approach to the congruence subgroup problem for the Teichmüller modular group along the lines of Boggi (Math Nach 279(9–10):953–987, 2006).

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.

The congruence subgroup property for the hyperelliptic modular group: the open surface case

Let $\cM_{g,n}$ and $\cH_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $\GG_{g,n}$ and $H_{g,n}$, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on $\cH_{g,n}$ defines a monomorphism $H_{g,n}\hookra\GG_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $\GG_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $\GG_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookra\out(\pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\ld$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $\pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\ra\out(\pi_1(S_{g,n})/K)$ is contained in $H^\ld$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.

Bak's work on theK-theory of rings

AbstractThis paper studies the work of Bak in algebra and (lower) algebraicK-theory and some later developments stimulated by them. We present an overview of his work in these areas, describe the setup and problems as well as the methods he introduced to attack these problems and state some of the crucial theorems. The aim is to analyse in detail some of his methods which are important and promising for further work in the subject. Among the topics covered are, unitary/general quadratic groups over form rings, structure theory and stability for such groups, quadraticK2and the quadratic Steinberg groups, nonstableK-theory and localisation-completion, intermediate subgroups, congruence subgroup problem, dimension theory and surgery theory.

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.

Profinite Teichmüller theory

AbstractFor 2g – 2 + n > 0, let Γg, n be the Teichmüller group of a compact Riemann surface of genus g with n points removed Sg, n , i.e., the group of homotopy classes of diffeomorphisms of Sg, n which preserve the orientation of Sg, n and a given order of its punctures. There is a natural faithful representation Γg, n → Out(π 1(Sg, n )). For any given finite index subgroup Γλ of Γg, n , the congruence subgroup problem asks whether there exists a finite index characteristic subgroup K of π 1(Sg, n ) such that the kernel of the induced representation Γg, n → Out(π 1(Sg, n )/K ) is contained in Γλ . The main result of the paper is an affirmative answer to this question. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)