Achievable ranks of intersections of finitely generated free groups
Richard P. Kent
我们回答一个由A引起的问题。Myasnikov通过证明所有期望的秩都是自由群的n-生成子群的交集的秩。
We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.
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The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.
Howson property and finitely generated intersection problem for monogenic inverse semigroups
研究单演逆半群作为半群时的豪森性质，除单演自由逆半群外都具有该性质，且对于此特殊情况，判定两个有限生成子半群的交是否有限生成的问题可算法求解。
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric s…
Intersection configurations in free and free times free-abelian groups
本文研究了自由群与自由乘自由阿贝尔群中的相交构型，证明了Howson性质是非交换自由群中的唯一障碍，并为FTFA群提供了判定多重相交有限生成性及计算基的算法，最终证明任何相交构型在足够大的FTFA群中均可实现。
Algorithmic Problems in Amalgams of Finite Groups: Conjugacy and Intersection Properties
Stallings提出的处理自由群的生成子群的几何方法被成功地用于解决自由群及其子群的判定问题。在本文中，我们采用作者发展的广义Stallings方法来解决有关有限群混合的有限生成子群的各种算法问题。
We show that the intersection of parabolic subgroups of an even finitely generated FC-type Artin group is again a parabolic subgroup.
The finitely generated intersection property in fundamental groups of graphs of groups
本文研究图群的fundamental groups是否满足有限生成交集性质，通过顶点群、边群双余集和图结构的性质，给出一般判据，并应用于局部近凸超几何群的图，证明其当且仅当不包含F₂×Z为子群时满足该性质。
We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann. This result was previously known only in the case where $A$ or $B$ is cyclic by the $w$-cycles theorem of Helfer and Wise and, independently, Louder and Wilton. In order to prove our main theorem, we introduce new results regarding the structure of fibre products of finite graphs. We work with a generalised definition of graph immersions so that our results…
Intersection-saturated groups without free subgroups
本研究提出交饱和群的新判据，据此找到包括Thompson群、Grigorchuk群在内的新实例，证明了无交换自由子群的有限表现交饱和群存在，解决了Delgado等人的问题。
Finitely generated Howson groups which are not strongly Howson
本文构造了有限生成的Howson群但不是强Howson群的例子（含或不含挠元），回答了Zhang和Zhao提出的问题。
Intersections of translated algebraic subtori
利用有限生成阿贝尔群与阿贝尔复代数约化群之间的经典对应，研究阿贝尔复代数约化群中平移子群的相交理论，特别强调代数环面中（挠）平移子环面的相交。
An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.
We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs and how to repair this lemma.
Subgroup and Coset Intersection in abelian-by-cyclic groups
本文研究无限群中的子群交与陪集交判定问题，通过将二者归约为 Shifted Monomial Membership 问题，证明其在有限生成 abelian-by-cyclic 群中可判定，并讨论推广到任意 metabelian 群的障碍。
Semigroup intersection problems in the Heisenberg groups
本文研究了Heisenberg群中的半群交集问题和轨道交集问题，证明了这两个问题在特定群结构下的可判定性。
The symmetry of intersection numbers in group theory
对于有限生成群的适当子群，定义了一个子群与另一个子群的相交数，并证明了这个相交数是对称的.我们也给出了这个数字的解释。
On the intersection of subgroups in free groups: echelon subgroups are inert
自由群$F$的子群$H$称为在$F$中惰性，如果对于每个$G < F$，$H$与$G$的交集的秩不大于$G$的秩。本文推广了已知的惰性子群族。证明了1-生成元自同态的惯性性质。自由群中的等差、梯子群是惰性的。梯队子群是通过一组生成元定义的，这些生成元相对于自由群的某个有序基呈梯队形式，并且可以被视为自由因子的推广。例如，n-生成的自由群的自同构的固定子群是梯队子群。证明主要遵循图论或组合方法。
Howson's property for semidirect products of semilattices by groups
逆半群$S$是Howson逆半群，如果$S$的逆生成逆子半群的交是逆生成的.给出半格$E$上的群$G$的局部有限作用$θ$，证明了$E \ast_θ G$是Howson逆半群当且仅当$G$是Howson群.它还表明，这种等价性失败的任意行动。
We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $τ\colon U \to K$ satisfying $τ(k) = k$ for every $k \in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products, and deduce that Howson's theorem holds for the Sylow subgroups of …
We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K of G, decides whether $H \cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H \cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian grou…
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.
We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, then the WN-coefficient $σ(H_1)$ of $H_1$ is rational and can be computed in exponential time in the size of $H_1$. This coefficient $σ(H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1 * G_2$, it is true that $\bar {\rm r} (H_1, H…
We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if $H_1$ is a finitely generated subgroup of a free group $F$, then the WN-coefficient $σ(H_1)$ of $H_1$ is rational and can be computed in deterministic exponential time in the size of $H_1$. This coefficient $σ(H_1)$ is the minimal nonnegative real number such that, for every finitely generated subgroup $H_2$ of $F$, it is true that $\bar {\rm r}(H_1, H_2) \le σ(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar{ {\rm r}} (H) := \max ( {\rm r} (H)-1,0)$ is…
An intersection functional on the space of subset currents on a free group
Kapovich和Nagnibeda在秩为$N\geq 2$的自由群$F_N$上引入了子集流的空间$\mathcal{S} {\rm Curr}(F_N)$，它可以被认为是$F_N$的所有共轭类的集合的测度论完备化.定义了$F_N$的两个双生成子群$H$和$K$的积$\mathcal{N} (H,K)$，并将积$HgK\ (g\in F_N)$推广为连续对称$\mathbb{R}_{\geq 0}$ -双线性泛函$\mathcal{N} \colon \mathcal{S} {\rm Curr} (F_N)\times \mathcal{S} {\rm Curr} (F_N)\to \mathbb {R}_{\geq 0}$ .我们还回答了Kapovich和Nagnibeda提出的一个问题。$\mathcal{N}$的定义起源于加强的汉娜·诺伊曼猜想，该猜想已经被Mineyev证明，并且可以表述如下：$\mathcal{N} (H,K)\leq \overline{\rm rk} (H) \overline{\rm rk} (K)$对$F_N$的任何生成子群$H$和$K$成立。作为我们定理的推论，这个不等式被推广到子集电流的不等式。
Splittings of Non-Finitely Generated Groups
在几何群论中，人们利用空间上的群作用来获得群的信息。一个自然使用的空间是一个群的凯莱图。我们遇到的凯莱图论证往往要求局部有限性，因此需要群的有限生成。在这篇文章中，我采用了有限生成群分裂的交集理论（由Scott，Scott-Swarup和Niblo-Sageev-Scott-Swarup提出），并对其进行了修改以去除有限生成假设。鉴于上述作者依赖于凯莱图的局部有限性，我利用Bass-Serre树进行分裂。
We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply …
The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.
The geometry of subgroups of mapping tori of free groups
本研究证明有限生成自由群的映射环面相对其典范极大子映射环面族是相对双曲且局部相对拟凸的，据此刻画了自由循环群与单关系群中的局部拟凸双曲群，并推广了Dehn函数计算等相关结果。
On the Howson property of descending HNN-extensions of groups
一个群$G$被称为具有Howson性质（或者是一个Howson群），如果$G$的任意两个双生成子群的交集是双生成子群。证明了在HNN-扩张的基本群所满足的某些假设下，降HNN-扩张不是Howson群。特别是，结果的文件加入了伯恩斯-布伦纳结果（收到于1979年）意味着任何下降HNN-扩展的非循环自由群没有豪森财产。
Graphs, free groups and the Hanna Neumann conjecture
给出了自由群的生成子群的交的秩的一个新的界，用拓扑术语表述，并且非常符合Stallings的精神。该界是对加强的汉娜·诺伊曼猜想的贡献（尽管不幸的是不是解决方案）。
Pullbacks and intersections in categories of graphs of groups
本文在图组范畴中发展了一个分类框架，重点研究拉回结构，并通过构造A-积来捕捉图组基本群子群的交集。
Homotopy bases and finite derivation type for Schutzenberger groups of monoids
给出了一个非拟现么半群及其同伦基，并给出了该么半群的任意Schutzenberger群，本文的主要结果给出了Schutzenberger群的同伦基及其表示.在Schutzenberger群G（H）的R类R'只有1/2个H类的情况下，存在H的乘法右逐点稳定子的元素s，使得在幺半群在其R类上的左作用下，s的R类的轨道与R'的逆轨道的交是有限的，则表示和同伦基的有限性保持不变。
We present a new algorithm deciding if the intersection of a quasiconvex subgroup of a negatively curved group with a conjugate is finite. We also give a short proof of decidability of the membership problem for quasiconvex subgroups of finitely generated groups with decidable word problem.
3-manifold groups are virtually residually p
给定一个素数$p$，如果一个群的$p$ -幂指数正规子群的交是平凡的，则称该群为剩余$p$。一个群称为虚剩余$p$，如果它有一个有限指数子群是剩余$p$。众所周知，特征为零的域上的非线性生成的线性群对于除了100个$p$之外的所有域几乎是剩余的$p$。特别地，双曲三维流形的基本群几乎是剩余的$p$。众所周知，三维流形的基本群是剩余有限的。在本文中，我们证明了这些结果的一个共同推广：每个3-流形群是几乎剩余的$p$，但许多$p$。这给出了证据的猜想（瑟斯顿），基本群的3流形是线性群。
The Hanna Neumann Conjecture and the rank of the join
HannaNeumann猜想给出了自由群的n-生成子群的交的一个界.我们探索了该结果的自然扩张，结果证明该结果仅在有限指数情况下为真，并提供了一般情况的反例。我们还看到，基于图的方法产生的随机子群的自由群开发的Bassino，Nicaud和Weil是非常适合于产生子群与非平凡的交叉。同样的方法，然后使用生成一个反例，以类似的猜想古兹曼。
Relative hyperbolicity and relative quasiconvexity for countable groups
为研究相对双曲群的相对拟凸子群奠定了基础。这些基础要求我们首先建立可数相对双曲群（不一定是双曲生成的）的相干理论。我们证明了Gromov，Osin和Bowditch的相对双曲性可数群的定义的等价性。然后利用相对双曲群上的各种自然几何给出了相对拟凸子群的几个等价定义。我们证明了每个相对拟凸子群本身是相对双曲的，并且两个相对拟凸子群的交集也是相对拟凸的。证明了非畸变子群是相对拟凸的，并计算了非畸变子群的畸变。
Groups with poly-context-free word problem
我们考虑一类群体的字问题是多上下文无关的，也就是说，一个交叉的许多上下文无关的语言。我们证明了任何群，它实际上是一个自由群的直积的一个可生成的子群有poly-context-free字问题，并猜想，反过来也成立。我们证明了我们的猜想几类可溶性群体，包括亚阿贝尔群和扭自由可溶性群体，目前的进展解决一般可溶性群体的猜想。一些用于证明语言不是多上下文无关的技术可能是独立的兴趣。
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications
完成了F猜想的证明。Bogomolov和F. Pop：设$F_{1}$和$F_{2}$是有限生成的域，分别在$k_{1}$和$k_{2}$上具有超越度$\geq 2$，其中$k_{1}$是$\bar{\mathbb{Q}}$或$\bar{\mathbb{F}}_{p}$，$k_{2}$是代数闭的。我们用$G_{F_1}$和$G_{F_2}$表示它们各自的绝对伽罗瓦群。则从$F_1$和$F_2$的不可分闭包的同构直到Frobenius扭到它们的伽罗瓦群的连续外同构的正则映射$φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$是双射。因此，在素域的代数闭包上的维数为$\geq 2$的簇的函数域是Anabel的。我们应用这一点给出了Grothendieck-Teichmüller群的元素成为$\bar{\mathbb{Q}}$的绝对Galois群的元素的一个充分必要条件。
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