M. R. Bridson, D. B. McReynolds, A. W. Reid, R. Spitler
We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups.
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We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first example of a non-orientable profinitely rigid hyperbolic 3-manifold.
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).
Profinite rigidity for two-bridge links and 3-tangle Montesinos links
本文证明了二桥链接和3-结蒙特斯林链接的补集基本群在紧致定向3流形的基本群中是profinite刚性的。
Profinite rigidity and quandle colourings
本文证明链环的有限quandle着色数等价于其外围系统的profinite完备化，将经典完备性问题转化为profinite刚性性，并发展了profinite quandle理论。
Profinite rigidity and geometric convergence
本文证明 profinitely rigid 有限体积双曲流形在几何拓扑下闭合并借助 bubble-drilling 构造推广到一大类带尖点双曲流形，涵盖 Whitehead link 与 Borromean ring 补空间。
Profinite rigidity of affine Coxeter groups
本文证明了仿射Coxeter群在 profinite 意义下是刚性的。
Profinite Non-Rigidity of Arithmetic Lattices and the Kähler Property
本文研究算术格的非有限刚性，刻画了允许非同构实形式的高秩李群绝对Dynkin类型，并证明有限呈现剩余有限群的Kähler性质不由其有限完备化决定。
Profinite rigidity of fibring
本文研究了有限呈现的LERF群的代数纤维化性质，证明其为profinite属性，并探讨了极限群乘积的纤维化及庞加莱对偶群的profinite刚性。
Profinite rigidity in lattices of $\mathrm{PSL}(2,\mathbb{C})$
该研究证明$\mathrm{PSL}(2,\mathbb{C})$中的所有格在同类格中具有投射有限刚性，且对其中任意格Γ，其投射有限完备化的外自同构群与自身外自同构群同构，推进了几何群论中投射有限刚性的研究。
Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds
本文通过研究带凹口的双曲3流形的Dehn填充，证明了某些流形在所有紧致定向3流形中是有限扩张刚性的，并改进了双曲结或双曲型卫星结的有限扩张刚性判据。
There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-$p$ completion of certain graph manifold groups.
Profinite almost rigidity in 3-manifolds
该研究证明带空或环面边界的紧致可定向3-流形具投射有限几乎刚性，结合外围结构可唯一确定混合3-流形Seifert部分同胚类型，否则无法唯一确定，证明基于JSJ分解。
A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients.
Groups of profinite type and profinite rigidity
本文研究 profinite 型群（可实现为 Galois 群的群）及其 profinite 刚性（唯一 profinite 拓扑），探讨阿贝尔群与群扩张的性质，并联系上同调比较映射的单射性与满射性。
We show that for a typical high rank arithmetic lattice $Γ$, there exist finite index subgroups $Γ_{1}$ and $Γ_{2}$ such that $Γ_{1} \not\simeq Γ_{2}$ while $\widehat{Γ_{1}} \simeq \widehat{Γ_{2}}$. But there are exceptions to that rule.
Let $Γ$ be a non-elementary Kleinian group and $H<Γ$ a finitely generated, proper subgroup. We prove that if $Γ$ has finite co-volume, then the profinite completions of $H$ and $Γ$ are not isomorphic. If $H$ has finite index in $Γ$, then there is a finite group onto which $H$ maps but $Γ$ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circleof ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We const…
We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $π_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $π_1(M)\cong F_r\rtimes_ψ\mathbb{Z}$ for some $ψ\in{\rm{Out}}(F_r)$.
Relative profinite rigidity of Baumslag-Solitar groups
本文研究Baumslag-Solitar群的相对有限完备刚性，证明剩余有限情形可仅凭有限商在同类群中被唯一识别。
The Profinite Rigidity of Free Metabelian Groups
本文证明有限生成自由亚交换群在绝对意义下具有投射刚性，综合运用诺特整环上模的投射刚性结果与 Groves–Miller 的同调刻画。
Profinite rigidity of crystallographic groups arising from Lie theory
本文证明由不可约根系产生的晶体群的有限直积具有 profinite 刚性（等价于一阶刚性），推广了仿射 Coxeter 群的结果，证明方法基于模型论。
Profinite rigidity and hyperbolic four-punctured sphere bundles over the circle
本文证明了圆周上的双曲四穿孔球丛可由其基本群的有限商群在所有3-流形群中被唯一区分，并通过改进Liu的结果证明了纤维拓扑类型可由profinite完备性检测。
If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $π_1M$ and $π_1N$ maps onto $G$ and the other does not.
Profinite rigidity of lamplighter groups
本文证明了当 p 为素数、n 为正整数时，灯台群 (Z/pZ)^n≀Z 具有投射有限刚性。
Smallest quotients and profinite rigidity of irreducible spherical type Artin groups
本文证明不可约球型Artin群在所有球型Artin群中具有profinite刚性，并计算了其最小非交换商，除特殊二面体情形外均与相应Coxeter群一致。
Profinite Rigidity over Noetherian Domains
本文研究诺特环上模的profinite刚性，建立基础命题，描述三个不变量，证明在特定条件下自由模和有限生成模的刚性，并应用于可解Baumslag-Solitar群，给出非阿贝尔单关系群和非LERF群中绝对profinite刚性的首批例子。
We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$, $F_4$, and $G_2$. In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\mathrm{SL}_{2n+1}(\mathbb{R})$, $\mathrm{SL}_{2n+1}(\mathbb{C})$, $\mathrm{SL}_n(\mathbb{H})$, or groups of type $E_6$.
Profinite rigidity of Kähler groups: Riemann surfaces and subdirect products
本文证明 Kähler 群可由 profinite 完备化在剩余有限 Kähler 群类中唯一确定，并由此推出非球面光滑射影簇由代数基本群同胚决定，主要工具是恢复全纯纤维化并证明 BNS 不变量的 profinite 不变性。
The Profinite Rigidity of Torsion-Free Lamplighter Groups
该研究证明了无 torsion 灯塔群在绝对意义上是 profinite 刚性的，其有限商唯一确定其同构类型。
Profinite rigidity of affine Coxeter groups
本文证明不可约仿射考克斯特群具有一阶刚性与绝对射有限刚性，其一阶理论无素模型，并证明有限秩泛考克斯特群及双曲单端直角考克斯特群是齐性的。
Lack of profinite rigidity among extensions with free quotient
本文构造了无穷多个互不同构的半直积，它们具有相同的profinite完备化，从而证明在自由商扩张中缺乏profinite刚性，并给出了自由群、自由阿贝尔群和曲面群的具体参数条件。
On profinite rigidity amongst free-by-cyclic groups I: the generic case
本文研究了自由-循环群的 profinite 刚性，证明了在特定条件下，超几何性、纤维秩和特征多项式等是 profinite 完成的不变量，并展示了一般自由-循环群的几乎刚性性质。
Profinite rigidity properties of central extensions of 2-orbifold groups
研究2-轨道群中央扩张的profinite刚性性质，证明刚性和非刚性现象并存，非刚性现象在扩展后消失。
Absolute profinite rigidity, direct products, and finite presentability
本文构造了有限展示的剩余有限群，证明其在有限展示群类中投射有限刚性但在有限生成群类中不刚性，并刻画了与刚性3-流形群直积投射有限等价的所有有限生成群。
Subgroup measures and profinite rigidity in free groups
本文研究了由自由群子群和元组诱导的度量，证明了子群的 profinite 刚性与元组的刚性等价，并推广了相关结果。
Virtual homology of limit groups and profinite rigidity of direct products
本文利用虚拟 Betti 数刻画与 pro-$p$ 群秩梯度技术，证明自由群和曲面群直积在有限展示剩余自由群中的有限完备刚性，并推出相关猜想与流形结论。
Profinite rigidity of simple closed curves in surface groups
研究闭可定向曲面上简单闭曲线，通过元素在到有限群满同态下的像特征，证明简单闭曲线集合在有限拓扑中是闭集，得到判定元素能否由简单闭曲线表示的新算法，还讨论了相关特殊情形。
Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity
本文为保测度等价关系的非周期包含引入代数不变量，借此证明了非顺从Baumslag-Solitar群直积自由pmp作用的稳定轨道等价刚性，并完成了其相对profinite完备化直积的完全分类。
On the profinite rigidity of free and surface groups
本文证明自由群或闭双曲曲面基本群 $S$ 的 pro-$p$ 完备化决定了其二生成子群的自由性，并证实 Remeslennikov 猜想在有限阿贝尔层次群类中成立，且 $S\times \mathbb{Z}^n$ 在剩余自由群中具有 profinite 刚性。
For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.
Profinite rigidity for free-by-cyclic groups with centre
本文证明了具有非平凡中心的自由-by-循环群及有限生成自由-by-(有限循环)群在具有相同有限商的条件下是profinite刚性的，并引入有限偏序集 fsc(G) 作为完全不变量，与曲面-by-循环群等缺乏刚性的情形形成对比。
Norm rigidity for arithmetic and profinite groups
本文证明具有有界初等生成的 ${\rm{SL}}_n(A)$ 的共轭不变范数具离散或预紧二分性，并将其与算术群及profinite群的刚性结果相关联，推导出对非剩余有限中心扩张近似的约束。
In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study various properties of $Comm(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $Comm(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Gri…
Relatively hyperbolic groups, Grothendieck pairs, and uncountable profinite ambiguity among fibre products
本文研究相对双曲群直积的投射有限刚性丧失问题，提出构造无限格罗滕迪克对和不可数互不同构子群族的一般方法，并以Weeks流形基本群和8字结补4重分支覆盖基本群为例，证明格罗滕迪克对完全解释了其直积的刚性丧失。
On profinite rigidity, Grothendieck pairs, and the second homology of some $3$-orbifold groups
本文开发了一种有效计算双曲有理同调$3$-球面orbifold群第二同调群的方法，并应用于Weeks流形及其相关格点，证明了这些格点的绝对profinite刚性，并构造了Grothendieck对。
By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.
We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in $\operatorname{Sp}(n,\mathbb{R})$, $G_{2(2)}$, $E_8(\mathbb{C})$, $F_4(\mathbb{C})$, and $G_2(\mathbb{C})$ are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.
We study equivalence relations $\mathcal R(Γ\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Γ\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(Γ\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(Λ\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgro…
Higman-Thompson groups and profinite properties of right-angled Coxeter groups
本文证明了右角Coxeter群在Coxeter群中具有profinite刚性，并展示了某些RACG具有无限profinite genus，同时推广了Higman关于Thompson群$V$的生成结果。
We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a ``rigidity'' approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmueller groups.
A note on words having the same image on finite groups
本文研究了有限生成自由群中词的弱 profinite 刚性，证明了原始词的幂及表面词的弱 profinitely 刚性。
Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure
本文研究结群的分支子群及其 profinite 和上同调结构，通过定义meridional inertia子群和全局分支子群，分析其在有限商、profinite 完成和上同调中的性质。
Some remarks on Grothendieck pairs
本文重新审视格罗滕迪克对的概念，探讨了 profinite 刚性与左/右格罗滕迪克刚性之间的关系，并证明了某些群的刚性特性。
Anti-Classification Results for Rigidity Conditions in Abelian and Nilpotent Groups
本文证明了刚性阿贝尔群、Hopfian群及co-Hopfian群在相应可数群空间中的完备共解析性，并解决了Thomas提出的两个公开问题，揭示了四类刚性条件的最高复杂度。
Prescribed Graphon Symmetries and Flavors of Rigidity
该研究将Frucht定理推广到图灵范畴，定义并证明了紧群图灵刚性与像刚性的等价性，明确了紧李群图灵刚性的判定条件。
Rigidity and Galois theory of 3-manifolds branched over Chebotarev links
本文在算术拓扑框架下，证明了3维流形的绝对伽罗华群决定其分支覆盖结构，类比数论中的Neukirch-Uchida定理。
Subgroups with all finite lifts isomorphic are conjugate
该研究证明有限群中所有有限提升同构的子群必为共轭，并回答了Dipendra Prasad关于余集等价子群未必同构的问题，同时探讨了与 profinite 刚性、anabelian 几何等领域的联系。
Center-freeness of finite-step solvable groups arising from anabelian geometry
研究双曲曲线的最大m步可解商的无中心性，证明其无挠且无中心，进而推导出m步可解Grothendieck猜想的刚性性质。
We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set gra…
The Manhattan curve, ergodic theory of topological flows and rigidity
本文为一般非初等双曲群引入曼哈顿曲线，利用拓扑流遍历理论和Patterson-Sullivan测度证明其可微性与刚性，并给出三角形群等显式计算。
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