Henry Wilton, Pavel Zalesskii
Let $M$ be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of $π_1(M)$ is efficient with respect to the JSJ decomposition of $M$. We go on to prove that $π_1(M)$ is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if $M$ is a graph manifold then $π_1(M)$ is conjugacy separable.
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In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all $n$-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of profinite completion of the fundamental group of $X$ in terms of the representation genus of its holonomy group.
$p$-Adic compactifications of geometric loop and diffeomorphism groups of compact manifolds on finite-dimensional spaces over non-Archimedean fields are investigated. Weakened topology is introduced. The structure of newly constructed compact groups is studied. Representations of such profinite groups are discused.
In this article we study the extent to which an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order.
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
Prosoluble subgroups of the profinite completion of the fundamental group of compact 3-manifolds
本文研究了 3 维流形群和几乎紧特殊群的 profinite 完备群的有限生成可解子群结构。
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.
On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the $A$-polynomial
该研究证明带尖点有限体积双曲3-流形基本群的投射有限完备化同构具正则与周边正则性，并得出S³中素纽结的A-多项式是投射有限不变量（可能差镜像）。
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。
Regularity of profinite isomorphisms of hyperbolic 3-manifolds
加强Liu的结果，证明双曲3 - 流形的有限完备同构在Boileau和Friedl意义下是正则的。
We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.
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。
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.
Profiniteness of higher rank volume
本文证明了高秩半单李群中不可约格的协体积由 profinite 完备化决定，并在无 CSP 条件下证明八元数双曲同余流形的体积是 profinite 不变量。
We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.
Let $(\mathcal{G},Γ)$ be an abstract graph of finite groups. If $Γ$ is finite, we can construct a profinite graph of groups in a natural way $(\hat{\mathcal{G}},Γ)$, where $\hat{\mathcal{G}}(m)$ is the profinite completion of $\mathcal{G}(m)$ for all $m \in Γ$. The main reason for this is that $Γ$ is finite, so it is already profinite. In this paper we deal with the infinite case, by constructing a profinite graph $\overlineΓ$ where $Γ$ is densely embedded and then defining a profinite graph of groups $(\widehat{\mathcal{G}},\overlineΓ)$. We also prove that the fundamental group $Π_1(\widehat{…
Let $G$ be 2-generated group. The generating graph $Γ(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a 2-generated profinite group $G,$ considering in this case topological generation. We prove that the set $V(G)$ of non-isolated vertices of $Γ(G)$ is closed in $G$ and that, if $G$ is prosoluble, then the graph $Δ(G)$ obtained from $Γ(G)$ by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinit…
We show that the quasiconvex subgroups in doubles of certain negatively curved groups are closed in the profinite topology. This allows us to construct the first known large family of hyperbolic 3-manifolds such that any finitely generated subgroup of the fundamental group of any member of the family is closed in the profinite topology.
Profinite properties of algebraically clean graphs of free groups
本文证明了代数干净的群图对每个素数 $p$ 具有几乎剩余 $p$-有限和上同调 $p$-完备性，且上同调良好，并应用于二维 Artin 群。
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.
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。
We define cofinite graphs and cofinite groupoids in a unified way that extends the notion of cofinite groups introduced by Hartley. The common underlying structure of all these objects is that they are directed graphs endowed with a certain type of uniform structure, that we call a cofinite uniformity. We begin by exploring the fundamental theory of cofinite directed graphs in full generality. The general theory turns out to be almost completely analogous to that of cofinite groups. For instance, every cofinite directed graph has a unique completion which is a com- pact cofinite direct graph. …
Profinite properties of residually free groups
该研究在剩余自由群的投射有限性质方向取得进展，证明了素三维流形基本群与$\mathbb{Z}^n$的直积在有限生成剩余自由群中投射有限刚性，还得到相关投射有限不变量并推广了已有引理。
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 $ψ\in{\rm{Out}}(F_r)$.
For a formation $\mathfrak{F}$ of finite groups, tight connections are established between the pro-$\mathfrak{F}$-topology of a finitely generated free group $F$ and the geometry of the Cayley graph $Γ(\hat{F_{\mathfrak{F}}})$ of the pro-$\mathfrak{F}$-completion $\hat{F_{\mathfrak {F}}}$ of $F$. For example, the Ribes--Zalesskii-Theorem is proved for the pro-$\mathfrak{F}$-topology of $F$ in case $Γ(\hat{F_{\mathfrak F}})$ is a tree-like graph. All these results are established by purely geometric proofs, without the use of inverse monoids which were indispensable in earlier papers, thereby g…
The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for…
We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $\hat{\mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for wh…
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。
The virtually generating graph of a profinite group
研究有限生成 profinite 群的虚拟生成图，证明移除孤立顶点后的图连通分支数可任意指定，且另一相关子图连通且直径至多 3。
An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements displaced by any non-identity automorphism. In this paper we confirm Tom Tucker's "Infinite Motion Conjecture" that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of a sequence of finite permutation groups with disjoint domains, viewed as a permutation gr…
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.
For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.
Cobordism, spin structures, and profinite completions
针对带Serre意义下好基本群的光滑闭连通非球面流形，证明基本群投射有限完备化同构可推出流形配边、签名模8相等且旋结构性质一致，还讨论了紧流形的类似结论。
In this paper we investigate the following general problem. Let $G$ be a group and let $i(G)$ be a property of $G$. Is there an integer $d$ such that $G$ contains a $d$-generated subgroup $H$ with $i(H)=i(G)$? Here we consider the case where $G$ is a profinite group and $H$ is a closed subgroup, extending earlier work of Lucchini and others on finite groups. For example, we prove that $d=3$ if $i(G)$ is the prime graph of $G$, which is best possible, and we show that $d=2$ if $i(G)$ is the exponent of a finitely generated prosupersolvable group $G$.
Profinite Subgroup Accessibility and Recognition of Amalgamated Factors
本文证明投射有限群的可达子群可实现为连续导子的核，并据此将投射有限完备化的融合积分裂回推出到抽象群，推广了关于自由因子的已有结果。
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…
Probabilistic properties of profinite groups
本文研究了profinite群在有限可解群或幂零群类下的$\mathfrak C$-positive性质，建立了等价条件并证明此类群为virtually prosoluble/nilpotent，同时给出了相关图的推论。
In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.
Taut polynomials from finite quotients of fibered hyperbolic 3-manifold groups
本文证明，当单值映射为完全穿孔时，纤维化双曲3-流形群的有限商群能检测Thurston范数球的纤维化面的taut多项式，并发展了扭曲多变量Alexander多项式的profinite不变性的一般框架。
In this paper, we prove that all finitely generated 3-manifold groups are Grothendieck rigid. More precisely, for any finitely generated 3-manifold group $G$ and any finitely generated proper subgroup $H<G$, we prove that the inclusion induced homomorphism $\widehat{i}:\widehat{H}\to \widehat{G}$ on profinite completions is not an isomorphism.
Finite covers and strict boundary slopes of cusped hyperbolic 3-manifolds
本研究证明了基本群投射有限完备化正则同构的两个带尖顶双曲3-流形，拥有相同的A-多项式且强检测边界斜率匹配，为相关流形的代数拓扑性质研究提供了新结论。
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。
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。
We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, $C'(1/6)$ one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy…
We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($τ$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in gene…
Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show that there exist two such surfaces if and only if M is neither a Z_2 homology solid torus nor a Z_2 homology cobordism between two tori. In particular, the exterior of a link with at least 3 components always contain two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite.
We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products. Furthermore, we develop the tools to study the analogous question in the pro-$p$ case. For a wide class of groups we show that the relevant cyclic subgroups - which are called $p$-isolated - are closed in the pro-$p$ topology of the graph product. In particular, we show that every $p$-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-$p$ topology, and we fully characterise such subgroups.
Determining knot groups by finite quotients
本文证明双曲2-桥纽结在所有紧3-流形中可由其纽结群的profinite完备化唯一确定。
Normalizers of lattices and isometry groups of arithmetic hyperbolic manifolds
该研究证明PSL(2,C)及PO(n,1)中算术格的子格正规化子性质，推导投射有限柔性格的集合特征，还证实有限群均可作为算术双曲流形的等距群，证明依赖格正规化子与子群增长理论。
Finite quotients of 3-manifold groups
本研究围绕3-流形群的有限商存在性问题展开，结合推广半特征理论的拓扑方法与概率方法，借助Dunfield-Thurston随机模型给出分布公式，通过群上同调完整解答相关问题，并首次通过矩构造新的随机群分布。
We show there exists a closed graph manifold $N$ and infinitely many non-separable, horizontal surfaces $\{S_{n} \to N\}_{n \in \mathbb{N}}$ such that there does not exist a quasi-isometry $π_1(N) \to π_1(N)$ taking $π_1(S_{n})$ to $π_1(S_{m})$ within a finite Hausdorff distance when $n \neq m$.
Ascending chains of free subgroups in closed hyperbolic and graph 3-manifold groups
通过图论和Stallings折叠方法，证明了闭曲面群及闭双曲和图3-流形基本群中常秩自由子群满足升链条件。
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
Linearity of Graph Manifold Groups
该研究完成了每个紧致3-流形群都是线性的证明，通过为闭非几何图流形群的有限指数子群构造GL₅(Q[u])中的忠实表示，得出相关结论。
Let M be a graph manifold. We show that π_1M is the fundamental group of a compact nonpositively curved cube complex if and only if M is chargeless. We also prove that in that case π_1M is virtually compact special.
Let $f\colon M\to N$ be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen showed that if $f$ induces a homology equivalence on all finite covers, then $f$ is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a $3$-manifold whose fundamental group is residually $p$ for every prime $p$. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. A…
Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.
We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.
Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is virtually the fundamental group of a special nonpositively curved cube complex. We provide a sufficient collection for graph manifolds with boundary thus proving that their fundamental groups are virtually special, and hence linear.
We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of graph-manifolds with boundary in terms of certain finite two-colored graphs. A corollary is the quasi-isometry classification of Artin groups whose presentation graphs are trees. In particular any two right-angled Artin groups whose presentation graphs are trees of diameter at least 3 are quasi-isometric, answering a question of Bestvina; further, this quasi-isometry …
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