M S Raghunathan
This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.
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In this paper we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on 3 strands.
In this paper we provide a sufficient condition for a subgroup of a free group not to have the almost congruence extension property (ACEP). We also show that any finitely generated subgroup of a free group satisfies some a generalization of ACEP.
We prove that the power subgroup $G_5^5$ and the commutator subgroup $G_5'$ of the Hecke group $G_5$ are not congruence.
In this paper we describe the profinite completion of the free solvable group on m generators of solvability length r>1. Then, we show that for m=r=2, the free metabelian group on two generators does not have the Congruence Subgroup Property.
Every sequence of orbifolds corresponding to pairwise non-conjugate congruence lattices in a higher rank semisimple group over local fields of zero characteristic is Benjamini--Schramm convergent to the universal cover.
We prove the centrality of the congruence kernel for the elementary subgroup of a Chevalley group G of rank >1 over an arbitrary noetherian ring R (under some minor restrictions on R if G is of type C_n or G_2).
Congruence subgroups of small Artin and Coxeter groups
本文研究小Artin群与Coxeter群的同余子群，基于Squier矩阵表示在特定取值下的整性，推广辫群同余子群理论，并确定低层级主同余子群。
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.
Multi-GGS-groups have the congruence subgroup property
本文将 GGS-groups 的同余子群性质推广至 multi-GGS-groups 族，指出除常向量定义者外均具此性质，并强调推广证明需新思路。
We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in terms of several families of generators. Some of our results are new even in the absolute case and were previously studied only for groups over finite fields.
A just-infinite iterated monodromy group without the congruence subgroup property
本研究证明多项式z²+i的迭代单值群具备刚无限、正则分支且无同余子群性质，给出了此类多项式迭代单值群的首个实例，并补充了该群相关核的信息。
For an epimorphism pi of the free group F_n onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of F_n for which pi*f = pi. This is called the standard congruence subgroup of Aut(F_n) associated to G and pi. In the case n = 2 we present formulas for the index of Gamma(G,pi) where G is abelian or dihedral. Moreover, we show that congruence subgroups associated to dihedral groups provide a family of subgroups of arbitrary large index in Aut(F_2) generated by a fixed number of elements. This implies that finite index subgroups of Aut(F_2) cannot be written as free product…
We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche-Wendt groups.
We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$, analyzing the subnormal structure of $H$, and the orbit-stabilizer problem for $H$. Effective computation with subgroups of $\mathrm{GL}(n,\mathbb{Z}_m)$ is vital to this work. All algorithms have been implemented in GAP.
We give a complete classification of all nonuniform congruences on orthogonal rook monoids and symplectic rook monoids. We find that there are four kinds of nonuniform congruences on the orthogonal rook monoids ${OR}_n$ for even $n\ne 4$, and we describe each kind of the congruences explicitly in terms of normal subgroups of maximal subgroups. We also find that if $n = 4$, there are six kinds of nonuniform congruences on ${OR}_4$, and we describe these congruences using both $\mathcal{H}$-relations and certain normal subgroups of some maximal subgroups. In contrast, we find that there is only …
The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for $Aut(F_2)$, the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asada's proof into group-theoretic language. This translation enables us to simplify many parts of Asada's original argument and prove a quantitative version of the congruence subgroup property for $Aut(F_2)$.
Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and $\U(\Z G)$ the group of units of $\Z G$. The Congruence Subgroup Problem for $\U(\Z G)$ is the problem of deciding if every subgroup of finite index of $\U(\Z G)$ contains a congruence subgroup, i.e. the kernel of the natural homomorphism $\U(\Z G) \rightarrow \U(\Z G/m\Z G)$ for some positive integer $m$. The congruence kernel of $\U(\Z G)$ is the kernel of the natural map from the completion of $\U(\Z G)$ with respect to the profinite topology to the completion with respect to the topology defined by the congruence subgroup…
We state and study the congruence subgroup problem for groups acting on rooted tree, 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 part…
We determine the normalisers of congruence subgroups of $G_5$.
Let F_n be the free group of rank n and let Aut^+(F_n) be its special automorphism group. For an epimorphism pi : F_n -> G of the free group F_n onto a finite group G we call Gamma^+(G,pi) = {f in Aut^+(F_n) | pi*f = pi} the standard congruence subgroup of Aut^+(F_n) associated to G and pi. In the case n = 2 we fully describe the abelianization of Gamma^+(G,pi) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Gamma^+(G,pi) < Aut^+(F_2) has infinite abelianization.
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 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 cl…
On the congruence subgroup property for GGS-groups
本文证明具有非常量定义向量的 GGS-群满足同余子群性质，为每个奇素数 $p$ 提供了丰富的有限生成、剩余有限、非挠群实例，并回答了 Barnea 的问题；同时证明常量定义向量的 GGS-群同余核无限且非分支群。
We give here a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.
It is known that the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$ has a finite generating set. In this paper, we give a finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$.
We use buildings and group extensions to compute lower bounds on the top Betti numbers for the cohomology of the level p congruence subgroups of SL(3,Z) and Sp(4,Z).
Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $\infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of $G$. In addition, for all but finitely many cases we evaluate $ngncs(G)$, the smallest index of a normal genuine non-co…
In this article we present an unpublished proof of W. Thurston that pure braid groups have the congruence subgroup property.
In this work, we construct fundamental domains for congruence subgroups of $SL_2(F_q[t])$ and $PGL_2(F_q[t])$. Our method uses Gekeler's description of the fundamental domains on the Bruhat- Tits tree $X = X_{q+1}$ in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.
The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by ut…
Principal congruence subgroups in the infinite rank case
本研究将1960年代关于整数一般线性群与特殊线性群的经典结果推广到无限生成自由阿贝尔群的自同构群，给出其正规生成元、极大正规子群分类、主同余子群正规生成元及Brenner梯关系的类比结论。
Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the unipotent radicals of two opposite parabolic subgroups P^+ and P^- of G. Assume that 2 is invertible in R if G is of type B_n,C_n,F_4,G_2 and 3 is invertible in R if G is of type G_2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism between the profinite completion of E(R) and the congruence completion of E(R) with …
Congruence Subgroup Property for nilpotent groups and subsurface subgroups of Mapping Class Groups
本文证明有限型曲面映射类群中两类子群满足同余子群性质，并顺带给出 Ben-Ezra–Lubotzky 定理的初等证明。
Congruence subgroups of braid groups and crystallographic quotients. Part II
本文接续前期研究，针对与辫群$B_3$相关的同余辫群下中心列展开分析，证明其对应商群具有几乎晶体学性质，推进了同余子群与晶体学辫群领域的研究。
Subgroup congruences for groups of prime power order
本文研究素数幂阶群中满足子群封闭性质的计数量，证明它们模 p 同余于 1 的等价性，并给出正规子群情形及 Burnside 定理的加强版本。
We consider a generalisation of the Basilica group to all odd primes: the $p$-Basilica groups acting on the $p$-adic tree. We show that the $p$-Basilica groups have the $p$-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the $p$-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the…
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…
The motivating question of this paper is roughly the following: given a group scheme $G$ over $\mathbb{Z}_p$, $p$ prime, with semisimple generic fiber $G_{\mathbb{Q}_p}$, how far are open subgroups of $G(\mathbb{Z}_p)$ from subgroups of the form $X(\mathbb{Z}_p)\mathbf{K}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and $\mathbf{K}_p(p^n)$ is the principal congruence subgroup $\operatorname{Ker} (G(\mathbb{Z}_p)\rightarrow G(\mathbb{Z}/p^n\mathbb{Z}))$? More precisely, we will show that for $G_{\mathbb{Q}_p}$ simply connected there exist constants $J\ge1$ and $\varepsilon>0$, depending only …
The congruence subgroup problem for a finitely generated group $Γ$ asks whether $\widehat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $Γ=F_{2}$ and $Φ_{2}$) and a new result for $Γ=Φ_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(Φ_{2}\right)=\hat{F}_ω$ when $Φ_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_ω$ is the free pr…
A congruence subgroup property for symmetric mapping class groups
本文研究了对称映射类群的同余子群性质，并证明了在特定条件下，torsion元素在共轭区分上的性质。
Congruence subgroups of braid groups and crystallographic quotients. Part I
本文为系列第一部分，通过引入同余辫群元素，建立了晶体学辫群与同余辫群商群间的同构关系。
The congruence subgroup problem for a finitely generated group $Γ$ asks whether the map $\hat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C\left(\mathbb{Z}^{n}\right)=\left\{ 1\right\}$ for every $n\geq3$, but $C\left(\mathbb{Z}^{2}\right)=\hat{F}_ω$, where $\hat{F}_ω$ is the free profinite group on countably many generators. Considering $Φ_{n}$, the free metabelian group on $n$ generators, it was also proven that…
Quotients of braid groups by their congruence subgroups
该研究解决了Dan Margalit提出的公开问题，确定了辫群同余子群在整一般线性群中的像，并利用辛同余子群刻画了辫群模同余子群的商群结构。
We construct two families of representations of the braid group $B_n$ by considering conjugation actions on congruence subgroups of $GL_{n-1}(Z[t^{\pm 1},q^{\pm 1}])$. We show that many of these representations are faithful modulo the center of $B_n$.
We extend some algebraic properties of the classical modular group SL_2(Z) to equivalent groups in the theory of Drinfeld modules, in particular properties which are important in the theory of modular curves. We study cusp amplitudes and the level of a (congruence) subgroup of SL_2(D) for any Dedekind domain D, as ideals of D. In particular, we extend a remarkable result of Larcher. We introduce finer notions of quasi-amplitude and quasi-level, which are not required to be ideals and encode more information about the subgroup. Our results also provide several new necessary conditions for a sub…
For semigroup $S$, a commutative congruence $σ_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in "Two cancellative commutative congruences and group diagrams", Semigroup Forum (2011) 82: 338-353. Here we demonstrate that when the semigroup is in fact a group $G$, then Orientable($G$) is the commutator subgroup $[G,G]$ and $ G / σ_{orient}$ is the abelian quotient group $G / [G,G]$.
The congruence subgroup property for mapping class groups and the residual finiteness of hyperbolic groups
本文在假设所有双曲群剩余有限的前提下，证明了有限型双曲曲面映射类群的同余子群性质，并由此推出profinite等价的双曲3-流形必为commensurable。
We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.
Let $R$ be a commutative ring that is free of rank $k$ as an abelian group, $p$ a prime, and $SL(n,R)$ the special linear group. We show that the Lie algebra associated to the filtration of $SL(n,R)$ by $p$-congruence subgroups is isomorphic to the tensor product $\mathfrak{sl}_n(R\otimes_{\Z}\Z/p)\otimes_{\F_p}t\F_p[t]$, the Lie algebra of polynomials with zero constant term and coefficients $n\times n$ traceless matrices with entries polynomials in $k$ variables over $\F_p$. We use the Lie algebra structure along with the Lyndon-Hochschild-Serre spectral sequence to compute the $d^2$ homolog…
Let K be a subgroup of the inhomogeneous Hecke group G_5 of finite index. Suppose that the geometric level of K is r. Then K is congruence if and only if K contains the principal congruence subgroup of level 2r.
Improved stability ranges in the homology of Torelli and congruence subgroups
通过表示论方法，改进了Torelli子群和同余子群在特定线性群作用下的第二同调群的中心稳定范围。
Powers of half-twists and congruence subgroups of braid groups
本文研究辫群同余子群$B_n[m]$与生成元幂正规闭包的关系，刻画前者含后者有限指数的条件并给出有限商群生成元，对无限指数情况证明商群含自由子群，还计算了相关交换化并构造新有限商群。
We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.
We give a method to describe all congruence images of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if $n=2$ then we compute all congruence images only modulo primes. We propose a separate method that works for all $n$ as long as $H$ contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.
We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\SL_2(\Z)$ that are projectively equivalent to one of the groups $Γ_0(N)$ or $Γ_1(N)$. A complete answer to this question is obtained: In case of $Γ_0(N)$ such noncongruence subgroups exist precisely if $N\not\in {3,4,8}$ and we additionally have either that $4\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $Γ_1(N)$ these noncongruence subgroups exist precisely if $N>4$. As in our previous paper t…
Generators for the level $m$ congruence subgroups of braid groups
本文证明辫群水平 $m$ 同余子群 $B_n[m]$ 由半扭转的 $m$ 次幂与辫 Torelli 群生成，解决 Margalit 问题并推广已有结果。
The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero, and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a "p-divided power algebra". The work is an extension of the work in arXiv:1204.5390v1 as well as extensions of a classica…
Congruence subgroups and crystallographic quotients of small Coxeter groups
本文研究小 Coxeter 群的全等子群性质，证明无限且非 virtually abelian 的小 Coxeter 群不满足该性质，并确定孪生群与三重群的晶体学商群及其维数和 holonomy 表示。
We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.
Connected fundamental domains for congruence subgroups
本文通过研究同余子群的项目线结构，构造了连通的基本域代表集，并引入了多重性函数M与可计算函数W，展示了其在数学中的应用。
Stable cohomology of congruence subgroups
研究同余子群SL_n(Z,p^m)在度数小于p-1时的F_p上同调，提出并验证了Calegari的公式，同时推导了SL_n(Z/p)在特定扭曲系数下的稳定上同调公式。
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