$Aut F_2$的同余子群性质：Asada定理的群论证明

低秩自由群与自由亚阿贝尔群的同余子群问题

The Congruence Subgroup Problem for low rank Free and Free Metabelian groups

一个群$Γ$的同余子群问题是问$\widehat{Aut\left(Γ\right)}\to Aut(\hatΓ)$是否是单射的，或者更一般地说，它的核$C\left(Γ\right)$是什么？这里$\hat{X}$表示$X$的profinite完成。本文首先给出了两个已知结果（关于$Γ=F_{2}$和$Φ_{2}$）的两个新的简短证明和关于$Γ=Φ_{3}$的一个新结果：1. $C\left(F_{2}\right)=\left\{ e\right\}$，当$F_{2}$是两个发生器上的空闲组时。2. $C\left(Φ_{2}\right)=\hat{F}_ω$当$Φ_{n}$是$n$生成元上的自由亚阿贝尔群，$\hat{F}_ω$是$\aleph_{0}$生成元上的自由profinite群时。3. $C\left(Φ_{3}\right)$含有$\hat{F}_ω$。结果2.和3.应该与第一作者即将得出的结果形成对比，该结果表明$C\left(Φ_{n}\right)$是$n\geq4$的阿贝尔。

根树自同构群的pro-$\mathcal{C}$同余性质

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

本文推广了作用于根树的群的同余子群问题，引入了基于有限群伪簇的pro-$\mathcal{C}$完备概念，给出了弱正则分支群满足$\mathcal{C}$-CSP的充要条件，并据此证明了Basilica群及常数定义向量GGS-groups具有$p$-CSP。

The Congruence Subgroup Problem for the Free Metabelian group on $n\geq4$ generators

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…

Drinfeld模群的真非同余子群

Genuine non-congruence subgroups of Drinfeld modular groups

设$A$是有限域$F_q$上的代数函数域$K$中的元素在固定位置$\infty$外积分的环。在较早的文件中，我们已经表明，德林费尔德模群$G=GL_2(A)$有自同构映射的同余子群非同余子群。在这里，我们证明了存在（不可数许多）正常真正的非同余子群，定义为那些保持非同余下的行动下的每一个自同构的$G$。此外，对于所有，但大多数情况下，我们评估$ngncs(G)$，正常真正的非全等子群的最小指数的$G$，并比较它的最小指数的任意正常非全等子群。

The IA-congruence kernel of high rank free Metabelian groups

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 this paper we investigate $C\left(IA(Φ_{n}),Φ_{n}\right)$, where $Φ_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(Φ_{n})=\ker(Aut(Φ_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(Φ_{n}),Φ_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happen…

CREAM: a Package to Compute [Auto, Endo, Iso, Mono, Epi]-morphisms, Congruences, Divisors and More for Algebras of Type $(2^n,1^n)$

The CREAM GAP package computes automorphisms, congruences, endomorphisms and subalgebras of algebras with an arbitrary number of binary and unary operations; it also decides if between two such algebras there exists a monomorphism, an epimorphism, an isomorphism or if one is a divisor of the other. Thus it finds those objects for almost all algebras used in practice (groups, quasigroups in their various signatures, semigroups possibly with many unary operations, fields, semi-rings, quandles, logic algebras, etc). As a one-size-fits-all package, it only relies on universal algebra theorems, wit…

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We fill this gap by proving that whenever $v \equiv 39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there exists a KTS on $v$ points having at least $v-3$ automorphisms. This is only one of the co…

Torelli与同余子群同调中稳定范围的改进

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。

同余子群第一同调的表征稳定性

Representation stability for the first homology of congruence subgroups

研究映射类群和自由群自同构群的同余子群的第一同调，以及Torelli群交换化的余不变模在有限域上的模表示序列，确定其合成因子和重数，并建立Church-Farb意义下的周期表征稳定性，应用于研究带level 2结构的曲线模空间上的平坦线丛。

Central stability for the homology of congruence subgroups and the second homology of Torelli groups

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.

剪刀自同构群 I：同调稳定性与 K-理论

Scissors automorphism groups I: Homological stability and K-theory

本文证明在温和条件下剪刀自同构群的同调独立于对象，可由剪刀同余 K-理论谱表达，提供群论解释与计算方法，并在经典情形及后续中得出同调计算与结构结果。