Super-exponential distortion of subgroups of CAT(-1) groups
Josh Barnard, Noel Brady, Pallavi Dani
我们构造了二维CAT（-1）群，其中包含具有任意迭代指数畸变的自由子群，并且畸变大于任何迭代指数畸变。
We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.
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In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is closely related to the separability of this subgroup.
We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Σ;\mathbb{Z})$.
We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups of $\CAT(0)$ groups introduced by Macura and introduce a collection of normal subgroups of right-angled Artin groups. These groups provide …
Relative Subgroup Growth and Subgroup Distortion
研究了有限生成群在有限生成群中的相对增长性，以及相应的嵌入偏差函数。我们探讨了哪些函数等价于非线性生成子群的相对增长函数和偏差函数。我们还研究了群嵌入的这两个渐近不变量之间的联系。本文给出了有限生成群上的长度函数可以推广到更大群上的长度函数的条件。
Subgroup Distortion in Wreath Products of Cyclic Groups
本文研究了子群畸变对圈积Awr Z的影响，其中A是环生成的阿贝尔.我们证明了AwrZ的每一个生成子群都有与某个多项式等价的偏差函数.此外，对A无穷大，且对任意多项式l^k，存在A wr Z的2-生成子群，其偏差函数等价于给定多项式。此外，任意圈积H wr G中元素长度的公式也很容易地表明群Z_2 wr Z^2有畸变子群，而点灯群Z_2 wr Z没有畸变（畸变生成）子群。
In this paper we propose cryptosystems based on subgroup distortion in hyperbolic groups. We also include concrete examples of hyperbolic groups as possible platforms.
There is a family of hyperbolic groups known as hyperbolic hydra which contain heavily distorted free subgroups. We prove the existence of Cannon--Thurston maps (that is, maps of the boundaries induced by subgroup inclusion) for these free subgroups. It is known that Cannon--Thurston maps between hyperbolic space boundaries can exist even in the presence of arbitrarily heavy (even non-recursive) distortion. The hyperbolic hydra examples show that Cannon--Thurston maps can exist even between hyperbolic group boundaries in the presence of arbitrarily heavy primitive recursive distortion.
We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for $\mathrm{Out}(F_n)$.
Lipschitz retraction and distortion for subgroups of Out(F_n)
给定秩为n的自由群F_n的自由因子A，刻画了Out（F_n）中稳定A的共轭类的子群在Out（F_n）中的畸变.我们还证明了Aut（F_{n-1}）在Aut（F_n）中的自然嵌入的象是无畸变的，F_n的任意自由分裂的共轭类在Out（F_n）中的稳定子是无畸变的，并刻画了F_n的任意自由因子系统的共轭类的稳定子何时畸变.在所有的非失真证明，我们证明了更强的声明，子群的问题是Lipschitz收缩。作为应用，我们确定了Out（F_n）和Aut（F_n）的Dehn函数和自动机性.
We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.
Fractional Isoperimetric Inequalities and subgroup distortion
证明了存在无穷多个非整数$r>2$使得某个非整数群的Dehn函数为$\simeq n^r$ .对于每个正有理数$s$，我们构造了一对双表示群$H\subset G$，使得$H$在$G$中的变形为$\simeq n^s$。对于每个$s\ge 1$，我们还构造了等径函数为$\simeq n^s$的二次表示群.
We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontriv…
Distortion in Free Nilpotent Groups
本文证明了一个群生成的自由幂零群幂零群F的子群是无畸变的当且仅当它是F中一个有限指标子群的收缩群。
Distortion of wreath products in some finitely presented groups
圈积如Z wr Z不是有限可表示的，但可以作为有限可表示群的子群出现。在这里，我们计算Z wr Z作为Thompson群F的子群和Baumslag亚交换群G的子群的畸变。我们发现Z wr Z在F中是不失真的，但在G中至少是指数失真的。
Conjugacy in fibre products, distortion, and the geometry of cyclic subgroups
研究无挠双曲群中纤维积的共轭问题复杂性，通过建立不等式，将纤维积的共轭长度函数与\(Q\)中循环子群几何等联系起来，为扩展相关函数库提供工具。
Parabolic subgroups of Coxeter groups acting by reflections on CAT(0) spaces
本文考虑CAT（0）空间$X$的上紧离散反射群$W$.然后，$W$成为Coxeter群。本文研究Davis-Moussong复形$Σ(W,S)$与CAT（0）空间$X$的相似性，并给出了Coxeter群$W$的抛物子群的极限集的几个相似结果.
Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston maps do not exist for infinite infinite-index normal CAT(0) subgroups with isolated flats in non-hyperbolic CAT(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have …
In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$
We construct 4-dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^m)$ for integers $n, m \geq 1$ and 6-dimensional CAT(0) groups containing finitely presented subgroups whose Dehn functions are $\exp^{(n)}(x^α)$ for integers $n \geq 1$ and $α$ dense in $[1,\infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.
We construct CAT(0) groups containing subgroups whose Dehn functions are given by $x^s$, for a dense set of numbers $s \in [2, \infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.
We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.
The rate of escape of random walks on polycyclic and metabelian groups
利用子群畸变来确定一类多环群上简单随机游动的逃逸率，并证明了逃逸率在生成集变化时不变.对于亚阿贝尔群，我们定义了一种更强的子群畸变形式，它适用于非非线性生成的子群。在此假设下，我们计算的逃逸率为某些随机游动的亚阿贝尔群通过比较的耗散阿贝尔沙堆倾倒。
In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…
Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups
本文构造了线性CAT(0)群的子群，其具有特定的有限性类型且不允许拟等距嵌入到一般线性群中，并推广了第二作者的相关纤维积构造。
Euler characteristics of p-subgroup categories
计算了有限群的p-子群范畴的欧拉示性数
In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting of diffeomorphisms which are isotopic to the identity on S does not contain any distorted elements. Moreover, we prove a weak Tits alternative for these groups.
The Tits alternative for CAT(0) cubical complexes
证明了作用于CAT（0）三次复形的群的一个Tits择一定理.也就是说，假设$G$是一个群，它的有限子群的阶有界。我们证明，如果$G$正确地作用在有限维CAT（0）三次复形上，则要么$G$包含秩为2的自由子群，要么$G$是有限生成的并且几乎是阿贝尔的。特别地，上述结论对有限维CAT（0）立方复形上任意自由作用的群$G$都成立。
Homotopy equivalences between p-subgroup categories
设p为素数，G为可被p整除的有限群. Quillen证明了G的非恒等p-子群的Brown偏序集与它的非恒等初等交换子群的子集同伦等价.我们证明了G的非恒等p-子群的融合范畴也有类似的结论。G的p-子群的其他范畴也被考虑。
Structures induced on transversals to a subgroup of a group and hypergroups over the group
在群的子群的横截上，利用群的二元运算，定义了结构映射。在此基础上，引入了群上的超群的概念，推广了群关于正规子群的同群的概念。群上的超群范畴包含作为子范畴的群的群范畴、线性空间范畴、域范畴。
We study the structure of the commensurator of a virtually abelian subgroup $H$ in $G$, where $G$ acts properly on a $\mathrm{CAT}(0)$ space $X$. When $X$ is a Hadamard manifold and $H$ is semisimple, we show that the commensurator of $H$ coincides with the normalizer of a finite index subgroup of $H$. When $X$ is a $\mathrm{CAT}(0)$ cube complex or a thick Euclidean building and the action of $G$ is cellular, we show that the commensurator of $H$ is an ascending union of normalizers of finite index subgroups of $H$. We explore several special cases where the results can be strengthened and we…
In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F \wr \mathbb{F}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L \le…
We construct families of $CAT(0)$ groups containing finitely presented groups whose divergence functions are of the form $r^α$ for a dense set of exponents $α\in [2,\infty)$ and $r^q\log(r)$ for integers $q \geq 2$. The same construction also yields examples of $CAT(0)$ groups containing contracting elements which are not contracting in certain finitely presented subgroups.
Existence and non-existence of bounded packing in CAT(0) spaces and Gromov hyperbolic spaces
本文的主要结果是：给定一个在CAT（0）空间$X$上按等距几何作用的群$G$和由$X$的秩1等距生成的$G$的循环子群$H$，$H$在$G$中有界填充.我们给出了这个结果的两个证明。第一种是Hamenstadt对秩- $1$等距的刻画。第二个证明直接来自Dahmani-Guirardel-Osin和Sisto的一些结果。然后利用Mihailova的构造，证明了两个自由群$\mathbb F_2\times \mathbb F_2$的直积的一个无界生成子群的存在性，回答了Hruska-Wise的一个问题.利用Wise的{ \em modified Rip构造}和Baumslag，Bridson，Miller和Short的{ \bf 1-2-3}定理，证明了CAT（0）群的无界填充子群的存在性.
We compute the relative divergence and the subgroup distortion of Bestvina-Brady subgroups. We also show that for each integer $n\geq 3$, there is a free subgroup of rank $n$ of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This result answers the question of Carr about the minimum rank $n$ such that some right-angled Artin group has a free subgroup of rank $n$ whose inclusion is not a quasi-isometric embedding. It is well-known that a right-angled Artin group $A_Γ$ is the fundamental group of a graph manifold whenever the defining graph $Γ$ is a tree. We sh…
Hyperbolic hydra
我们给出了双曲群的例子，有限秩自由子群的巨大（阿克曼）失真。
Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups
我们引入了扰动直角Artin群类。这些是通过将Bieri双群粘合到标准直角Artin群中来构造的。作为第一个应用程序的建设，我们获得家庭的CAT（0）群，其中包含非$\mathrm{FP}_3$型，并有指数，或多项式Dehn函数的规定程度的非线性子群。
We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but not $F_{n+1}$ with no free abelian subgroups of rank greater than $\lceil\frac{n}{3}\rceil$.
Fractional distortion in hyperbolic groups
本文构造了双曲群$G$，使其包含任意非初等无挠双曲子群$H$，且$H$在$G$中的扭曲函数按迭代指数函数$\n\nexp^k(n^{p/q})$增长。
Conditions on the monodromy for a surface group extension to be CAT(0)
该研究将Llosa Isenrich和Py关于曲面-曲面群为CAT(0)群的单值性必要条件推广到更一般情形，证明含正规曲面子群且商群满足特定性质的CAT(0)群的无限单值性表示核有限，且柱形双曲群满足该性质。
Geometric density for invariant random subgroups of groups acting on CAT(0) spaces
我们证明了在CAT（0）空间上具有几何稠密作用的群的IRS也具有几何稠密作用;假设该空间是有限伸缩维数的或具有有限维Tits边界的局部紧的。这可以被认为是IRSs的Borel密度定理。
In this paper, we study distortion in the group $\mathcal A$ of Affine Interval Exchange Transformations (AIET). We prove that any distorted element $f$ of $\mathcal A$, has an iterate $f^ k$ that is conjugate by an element of $\mathcal A$ to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences we prove that no Baumslag-Solitar group, $BS(m,n)$ with $\vert m \vert \neq \vert n \vert$, acts faithfully by elements of $\mathcal A$, every finitely generated nilpotent group of $\mathcal A$ is virtually abelian and there is no distortion element in $\mat…
Accessible CAT$(-1)$ groups of critical exponent less than one
该研究针对可及CAT(-1)群，证明临界指数小于1时群的结构性质，给出相关推论，覆盖更大类群并补充JSJ相关结论。
Morse theory and conjugacy classes of finite subgroups
我们构造了一个CAT（0）群，它包含一个具有无穷多个有限阶元共轭类的非线性表示子群。与前面的例子（基于直角Artin群）不同，我们的环境CAT（0）群不包含任何秩为3的自由阿贝尔子群。我们还构造了具有无穷多个有限阶元素共轭类的映射类群、Aut（F）和Out（F）中的F_n型群的例子。
On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups
本文讨论了Coxeter群范畴中非标准子群的一种构造。给出了这类子群的增长级数的两个公式。作为应用，我们构造了一个旗形单凸多胞体，其f-多项式具有非实根.
The main result of this paper is that the categories of (right) hypergroups over the group and of triples, consisting of a group, its subgroup and a (right) transversal to this subgroup, are equivalent.
Commensurators of abelian subgroups of biautomatic groups
本文研究双自动群中有限生成阿贝尔子群的公度子，证明其必中心化有限指数子群，并据此给出非双自动群子群的 CAT(0) 群首例。
On the orbit category on nontrivial $p$-subgroups and endotrivial modules
本文确定了一大类有限群（包括具有亚循环 Sylow $p$-子群的奇数阶情形）的非平凡 $p$-子群轨道范畴的基本群，并借助 Grodal 方法描述了内平凡模群，改进了 Carlson 与 Thévenaz 的已知结果。
The question which motivates the article is the following: given a group acting on a CAT(0) cube complex, how can we prove that it is acylindrically hyperbolic? Keeping this goal in mind, we show a weak acylindricity of the action on the contact graph associated to a CAT(0) cube complex, and we prove a characterisation of WPD contracting isometries of CAT(0) cube complexes. As a first consequence, we find alternative arguments to show several criteria which were proved by Indira Chatterji and Alexandre Martin. Next, we show that if a group acts essentially and acylindrically on the hyperplanes…
Helly-type theorems, CAT$(0)$ spaces, and actions of automorphism groups of free groups
本文证明了群作用于CAT$(0)$空间的不动点定理，通过自举技术与Helly型定理得出几何感兴趣的群无全局不动点作用的最小维度下界，并证明自由群自同构群满足${\rm{FixDim}}({\rm{Aut}}(F_n)) \ge \lfloor 2n/3\rfloor$。
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。
CAT(0) spaces with boundary the join of two Cantor sets
我们将证明，如果一个真完备CAT（0）空间X有一个与两个康托集的并同胚的可视边界，并且X允许一个包含一个与Z^2同构的子群的群的几何群作用，则它的Tits边界是两个不可数离散集的球面并。若X是测地完备的，则X是一个积，并且这个群有一个有限指数子群同构于两个有界价丛树的等距群的积中的格。
Higher limits via subgroup complexes
研究函子F的逆极限的高阶导出函子子：D --> Z_{(p)} -mod，其中D是研究有限群G的分类空间的同伦理论时出现的标准范畴之一，例如，G的轨道范畴或Quillen范畴。这些更高的限制是重要的，例如，用于研究分类空间之间的映射以及群上同调。我们证明了这些上限可以用G中p-子群的子群复形的G-等变Bredon上同调来确定（即，G中p-子群的偏序集的神经，其值在G-局部系数系统中。我们研究何时可以使用较小的复合物，例如，仅取p-根式子群、p-中心子群、初等阿贝尔p-子群或其各种子集。由于子群复形是有限复形，而且通常相当小，这为这些上限提供了具体的、可计算的公式，推广了特别是Jackowski-McClure-Oliver的早期工作。作为一个应用程序，我们看看在特殊情况下，所有的上限消失，例如是情况下的群上同调。若F是轨道范畴上的函子，则我们的上限公式给出了F（G）在真子群上的五种不同的表达式。其中两个是“经典”即韦伯的确切序列麦基函子和公式计算稳定的元素，以前获得使用Alperin的融合定理。检查这种情况下也导致改进的清晰度结果的同源分解由于德怀尔和其他人。
On minimal epimorphic subgroups in simple algebraic groups of rank $2$
本文研究了秩 $2$ 单代数群中的极小满射子群问题，证明了在代数闭域上其闭满射子群的极小维数为 $3$。
On quasimorphisms and distortion in homeomorphism groups
本文研究了同胚群中的拟模与扭曲问题，识别了可扩展的拟模并探讨了其在群同调中的应用。
Pro-p groups with constant generating number on open subgroups
设p为素数。对G的所有开子群H满足d（H）= d（G）的n-生成pro-p群G进行分类.这里d（H）表示子群H的最小拓扑生成元数。在p-adic解析亲p群的范畴内，这回答了岩泽的一个问题。
Morse theory and conjugacy classes of finite subgroups II
本文构造了一个双曲群，它具有无穷多个有限阶元的共轭类。我们还使用了一个版本的莫尔斯理论与高维水平细胞和使用处理取消参数产生的CAT（0）群的子群的其他例子，无限多共轭类的有限阶元素。
Actions on CAT(-1) spaces with critical exponent less than 1
该研究针对在CAT(-1)空间上作用的离散等距子群，证明了临界指数小于1时其与极限集豪斯多夫维数的关系，得出极限集为康托集的结论，并应用该结果回答了Kapovich关于几何有限自由子群的问题。
Distortion element in the automorphism group of a full shift
本文证明了全移位自同构群的有限生成子群中存在字范数呈多对数增长的无限阶扭曲元素，并以此推导出子移位熵维数下界及sofic shift自同构群的性质，揭示了Turing机群与高维Brin-Thompson群的相关特征。
Hydra groups
我们给出的例子CAT（0），双自动，自由的循环，一个relator组有有限秩自由子群的巨大（阿克曼）失真。这就引出了德恩函数也是同样奢侈的群的基本例子。这种行为起源于字符串重写中赫拉克勒斯与九头蛇之战的表现。
Parabolic subgroups of Garside groups II: ribbons
我们引入并研究了与Garside群相关的带状广群。在一个技术假设下，我们证明了这个范畴是一个Garside广群。我们把这个广群分解成它的两个抛物子广群的半直积，并给出一个广群表示。为了建立后一个结果，我们描述了Garside群中的拟中心化子。所有的结果举行的特殊情况下，Artin-Tits集团的球型。
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