The Euler characteristic of the Whitehead automorphism group of a free product
Craig Jensen, Jon McCammond, John Meier
建立了标号超树格上的一个组合求和恒等式，使人们能够获得关于群的自由积的各种自同构群的欧拉示性数的具体信息.
A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups.
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In analogy with the free factors of a free group we define special factors of Generalized Baumslag-Solitar (GBS) groups as non-cyclic subgroups which appear in splittings over infinite cyclic groups. We give an algorithm which, given a GBS group $G$ and an element $g \in G$, decides whether there exists a special factor $H < G$ such that $g \in H$. This algorithm is analogous to an algorithm by Whitehead for free groups. Furthermore we prove that given $g \in G$ there exists a unique minimal special factor containing $g$ and give an algorithm which finds it.
We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "train track directed" random walk of Kapovich-Pfaff. The tools developed here also allow one to construct a whole new array of ideal Whitehead graphs achieved by ageometric fully irreducible outer automorphisms in all ranks.
Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs
H. Masur和J. Smillie精确地证明了哪些奇异性指数列表来自伪Anosov映射类。在寻找自由群的外自同构的类似定理时，Handel和Mosher提出了一个问题：在Out（F_r）中，每个连通的单纯的（2 r-1）-顶点图是完全不可约外自同构的理想Whitehead图吗？我们回答这个问题的负面展示，对于每个r，连接（2 r-1）-顶点图的例子，是不是理想的Whitehead图的任何完全不可约的外自同构出（F_r）？在我们的证明过程中，我们还开发了用于“构造和分类自由群的完全不可约外自同构“的机器，以充分回答秩三情况下的问题。
Whitehead problems for words in $\ZZ^m \times F_n$
我们解决了怀特黑德问题的自同构，单态和自同态$\ZZ^m \times F_n$后，给出了明确的描述，这些家庭的转换。
Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups
本文的主要定理在Out（F_r）理论的背景下模拟了一个映射类群定理（H. Masur和J. Smillie），它精确地确定了哪些索引列表来自伪Anosov映射类。由于理想的怀特海图给出了一个更好的不变量在类似的设置一个完全不可约的外自同构，我们而是专注于确定哪些21个连接，无环，5顶点图是理想的怀特海图的年龄，完全不可约的外自同构的自由群的秩3。我们的主要定理通过证明恰好有18个图像这样产生来实现这一点。我们还给出了一种方法，用于确定某些并发症称为周期尼尔森路径，证明了存在方便分解的代表年龄，完全不可约外自同构具有连接，无环，（2 r-1）-顶点理想Whitehead图，并证明了一个标准，用于确定代表年龄，完全不可约外自同构。我们用来构造自由群的完全不可约外自同构的方法，以及我们的识别和分解技巧，可以用来扩展我们的主要定理，因为它们在任何秩中都是有效的。我们的证明方法主要依赖于Bestvina-Feigh-Handel训练轨道理论和吸引叠层理论。
J.H.C. Whitehead's second free-group algorithm determines whether or not two given elements of a free group lie in the same orbit of the automorphism group of the free group. The algorithm involves certain connected graphs, and Whitehead used three-manifold models to prove their connectedness; later, Rapaport and Higgins & Lyndon gave group-theoretic proofs. Combined work of Gersten, Stallings, and Hoare showed that the three-manifold models may be viewed as graphs. We give the direct translation of Whitehead's topological argument into the language of graph theory.
The object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in the study of free groups, and use the solutions of those problems to motivate topological/geometric tools. We do not aim to write down proofs which minimize the number of alphanumeric characters. We instead strive to write down proofs which maximize the development of broadly applicable geometric tools. In Part I we study problems solved by Nielsen and Whitehe…
Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2
该研究针对秩2自由群的Whitehead自同构问题，先证明轨道阻塞字的存在性，进而提出常数平均情况复杂度的判定算法。
Generators for the Pure Symmetric Outer Automorphisms of a Free Product
本文通过构造自由积外空间的良好子复形，几何地证明了给定分裂的纯对称外自同构可由因子外自同构与相对Whitehead外自同构生成。
We prove that a "random" free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a nongeometric $\mathbb R$-tree all of whose branch points are trivalent
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case complexity (i.e., the expected runtime) of algorithms that solve a well-known problem, the Whitehead problem in a free group, which is: given two elements of a free group, find out whether there is an automorphism that takes one element to the other. First we address a special case of the Whitehead problem, namely deciding if a given element of a free group is…
Peak reduction and finite presentations for automorphism groups of right-angled Artin groups
我们将自由群的峰值缩减算法（Whitehead定理）推广到关于一般直角Artin群A_Gamma的定理。作为一个应用，我们找到了自同构群Aut A_Gamma的一个有限表示，它推广了有限秩自由群的自同构群的McCool表示。我们还考虑了峰降的一个更强的推广，给出了一个反例，并证明了一个特例。
Generic properties of Whitehead's Algorithm and isomorphism rigidity of random one-relator groups
证明了求解固定自由群$F_k$中自同构问题的Whitehead算法具有强线性时间一般情况复杂度。这是通过显示算法的“硬”部分在线性时间内终止于一组指数通用的输入对来完成的。然后，我们将这些结果应用到一个关系群。我们得到了随机单关系群的Mostow型同构刚性结果：如果两个这样的群同构，则它们在\emph{given generating sets}上的Cayley图是等距的.虽然以前没有非平凡的例子，我们证明了单关系群是一般的\emph{complete}群，即它们有平凡的中心和平凡的外自同构群。我们还证明了在$Aut(F_k)$中$F_k$的类元的稳定子是由内自同构生成的循环群，并且证明了$Aut(F_k)$ -轨道在其增长熵意义下是一致小的.进一步证明了$k$生成元单关系子群的\emph{isomorphism types}的个数$I_k(n)$满足\[ \frac{c_1}{n} (2k-1)^n \le I_k(n)\le \frac{c_2}{n} (2k-1)^n, \]，其中$c_1=c_1(k)>0, c_2=c_2(k)>0$是与$n$无关的常数.因此，$I_k(n)$基本上以与长度为$n$的循环字的数目相同的方式增长。
A Bound for Higher Degree Automorphisms of $F_n$
本文利用 Whitehead 模型，将自由群自同构复合次数为 0 的界推广至至少一个自同构次数为 0 的情形。
Folding free-group automorphisms
我们描述了一个算法，使用Stallings的折叠技术分解的元素$Aut(F_n)$作为一个产品的Whitehead自同构（因此作为一个产品的尼尔森变换）。我们用这个给一个替代的方法找到一个有限的生成集的子群的$Aut(F_n)$，固定一个子集$Y$的基本元素，和子群，固定每个元素的$Y$共轭。我们表明，这后一个子群与$IA_n$的交集也是随机生成的。
Let $ϕ$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_ϕ = F_n \rtimes_ϕ \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite graph $K_{3,3}$ into $\partial G_ϕ$. To do so, we define the directional Whitehead graph and prove that an indecomposable $F_n$-tree is Levitt type if and only if one of its directional Whitehead graphs contains more than one edge. As an application, we obtain a direct proof of Kapovich-Kleiner's theorem that $\partial G_ϕ$ is homeomorphic to the Menger curve if the auto…
We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.
Automorphic orbits in free groups
设$F_n$是有限秩$n$的自由群。我们研究轨道$Orb_ϕ(u)$，其中$u$是一个元素的组$F_n$，下的作用下的自同构$ϕ$。如果这样一个轨道是有限的，我们精确地确定它的基数是多少，如果$u$贯穿整个群$F_n$，而$ϕ$贯穿整个群$Aut(F_n)$。另一个问题，我们在这里解决的是有关怀特海的算法，确定是否或不一个给定的元素的自由群有限秩是一个自守形象的另一个给定的元素。众所周知，该算法的第一部分（通过初等Whitehead自同构将给定的自由词减少到最小可能长度的自由词）是快速的（相对于词的长度的二次时间）。另一方面，算法的第二部分（应用于最小长度相同的两个单词）总是被认为非常慢。本文给出了第二部分的一个改进算法，并且我们相信这个算法总是在关于字长的多项式时间内终止。我们证明了当自由群秩为2时，这确实是一种情形。
We define and discuss a notion called fibered commensurability of outer automorphisms of free groups. This notion lets us study symmetry of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping classes is a commensurability invariant. One of the important facts of fibered commensurability on mapping class groups is for the case of pseudo-Anosovs, there is a unique minimal element in each fibered commensurability class. For outer automorphisms, we first show that being atoroidal and fully i…
Unipotent linear suspensions of free groups
受自由群外自同构共轭问题驱动，本文发展了由幂单线性增长自同构产生的 free-by-cyclic 群的算法理论，计算了其规范分裂与自同构群，证明了有效凝聚性，解决了混合 Whitehead 问题，并最终应用于解决该类自由群外自同构的共轭问题。
Arc-transitive maps with edge number coprime to the Euler characteristic -- I
本文作为边数与欧拉示性数互素的边传递地图分类系列论文之一，建立分类框架，完成可解自同构群下弧传递地图的分类，揭示边数对欧拉示性数的影响，涉及多种新颖弧正则地图的构造。
Arc-transitive maps with coprime Euler characteristic and edge number -- II
作为系列论文第二篇，本文完成不可解自同构群的弧传递地图分类，结合首篇成果完整刻画欧拉示性数与边数互素的弧传递地图，分类过程构建了若干新颖反向地图。
The cogrowth inequality from Whitehead's algorithm
本文基于Whitehead算法改进，研究自由因子H的共增长不等式，通过自动机构造和Perron-Frobenius理论证明H的共增长小于phi(H)的共增长。
On Bi-rotary Maps of Negative Prime Power Euler Characteristic
本文研究双旋转地图的自同构群结构，通过引入双定向与弧正则子群的概念，刻画了欧拉示性数为负素数幂的双旋转地图的自同构群结构。
A classification of regular maps with Euler characteristic $-pq$
本文对具有欧拉特征数$-pq$的正则映射进行了分类，并指出对于孪生素数$p$和$q$，存在三个具有可解自同构群的正则映射。
Finite groups of untwisted outer automorphisms of RAAGs
本文证明直角 Artin 群外自同构群中由 Whitehead 自同构和反演生成的子群的任何有限子群都固定可缩空间中的一点，从而推广了 Outer Space 的相应结果并推出有限子群共轭类有限。
The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of $m,n$. We also study periodic points of endomorphisms of $Z^m\times F_n$, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of $Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real eleme…
A note on test elements for monomorphisms of free groups
本文通过利用Whitehead图和自由群在其Caley图上的作用，提出了一种几何条件，用于构造自由群单射的测试元素。
Automorphic orbits in free groups: recent progress
本文综述了自由群中各类自守轨道的渐近性质的最新进展，包括给定长度的潜在正元素计数、Whitehead自守问题的复杂度以及轨道阻塞词等。
We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…
Distance in the Ellipticity Graph
定义了自由群$F$的椭圆度图. Kapovich和M. Lustig为了研究作用在此图上的$F$的外自同构群而提出了一种新的方法。该图被构造成类似于曲面的曲线复合体。它是一个二部图，其顶点是$F$的非平凡元素的共轭类和形式为$F=A*B$的真自由积分解的等价类。只要共轭类在$A$或$B$中有代表，共轭类就通过边连接到自由积分解$A*B$.本文使用Stallings子群$X$ -有向图和Whitehead自同构构造算法，确定当两个顶点之间的椭圆度图的距离是2，这两种类型的顶点。
Non-orientable regular maps with negative prime-power Euler characteristic
本文对欧拉示性数为负奇素数幂的曲面上的正则地图进行分类，通过分析自同构群商群结构，得到18个非空正则地图族。
A classification of vertex-reversing maps with Euler characteristic coprime to the edge number
本文对边数与欧拉特征互质的可解顶点翻转图进行了分类，发现其包含三个族，并基于有限几乎索引循环群的特征化进行了证明。
Twisted conjugacy classes for polyfree groups
设$G$是一个无环生成的多角自由群.若$G$具有非零的欧拉示性数线，则证明了$Aut(G)$有一个有限指数子群，其中每个自同构都有无穷多个Reidemeister数.对于某些长度为2的$G$，我们证明了每个自同构的Reidemeister类的数目是无限的.
Outer automorphism groups of hyperbolic groups, bounded extensions, and hierarchical hyperbolicity
本文证明在JSJ分解的适度可定向条件下，一端双曲群的外自同构群实质上是一个层次双曲群，通过证明有限指数子群是轨道流形映射类群乘积的中心扩张且扩张具有有界欧拉类，并给出反例说明完全外自同构群不一定是HHG。
Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler cha…
In this note we study the problem of characterizing the complex affine space $\mathbb{A}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\mathrm{Aut}(X)$ and $\mathrm{Aut}(\mathbb{A}^n)$ are isomorphic as abstract groups. If $X$ is either quasi-affine and toric or $X$ is smooth with Euler characteristic $χ(X) \neq 0$ and finite Picard group $\mathrm{Pic}(X)$, then $X$ is isomorphic to $\mathbb{A}^n$. The main ingredient is the following result. Let $X$ be a smooth irreducible quasi-projective variety of dimens…
The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the rational primes. We manufacture the first example of a torsion-free finitely generated nilpotent group G such that the local Euler factors of its pro-isomorphic zeta function do not satisfy functional equations. The group G has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable algebra…
The pro-isomorphic zeta function of a finitely generated nilpotent group $Γ$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $Γ$. Such zeta functions can be expressed as Euler products of $p$-adic integrals over the $p$-adic points of an algebraic automorphism group associated to $Γ$. In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-$2$ nilpotent groups; these groups can be viewed as gene…
The Euler characteristic of the moduli space of graphs
本文研究了秩 $n$ 图模空间、自由群外自同构群与 Kontsevich 李图复形具有相同有理上同调，证明其欧拉示性数随 $n$ 趋于无穷大的渐近增长规律，从而得出该上同调总维数快速增长的结论。
A classification of regular maps with Euler characteristic $-p^4$ for a prime $p\geq 5$
本文研究了欧拉特征数为$-p^4$的正则图，证明了其Sylow $p$-子群的阶数限制，并分类了所有此类正则图，表明当$p$不在{2,3,5,7,13}时，相关曲面不支持正则图。
On endomorphisms of torsion-free hyperbolic groups
设$H$是关于有限生成集$S$的无挠的$δ$ -双曲群.令$a_1,..., a_n$和$a_{1*},..., a_{n*}$是$H$的元素，使得对于每个$i=1,..., n$，$a_{i*}$与$a_i$共轭。然后，当且仅当对于$n$变量中的每个单词$W$，并且长度达到仅取决于$δ$、$\sharp{S}$和$\sum_{i=1}^n |a_i|$的可计算常数，$W(a_{1*},..., a_{n*})$与$W(a_1,..., a_n)$共轭时，存在统一共轭子。作为推论，我们推导出存在一个可计算常数$\mathcal{C}=\mathcal{C}(δ, \sharp S)$，使得对于$H$的任何自同态$ϕ$，如果$ϕ(h)$与$h$共轭，对于每个长度为$\mathcal {C}$的元素$h\in H$，则$ϕ$是内自同构。另一个推论是：如果$H$是一个无挠的共轭可分双曲群，则$\text{\rm Out}(H)$是剩余有限的。当特殊化的主要结果的情况下，自由群，我们得到一个混合版本的经典怀特海的算法的解决方案。
A Fixed Point Theorem for Deformation Spaces of G-trees
对于秩至少为2的自由群F_n，Out（F_n）的任意有限子群都可以实现为具有基本群F_n的图的自同构自同构群群. Zimmermann，Culler和Khramtsov证明了这个结果，称之为Out（F_n）实现。这一定理与Kerckhoff证明的Nielsen实现类似：对于具有负欧拉示性数的闭曲面，映射类群的任何有限子群都可以实现为双曲曲面的等距群。这两个定理都有重述方面的不动点的行动空间自然与他们。对于非负整数n，我们定义了一类群（GVP（n）），并证明了它们的外自同构群的一个类似的陈述。
Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces
设G是有限群，M是G-流形.我们引入了M/G的广义轨道不变量的概念，它与任意群Gamma，任意Gamma-集和连通流形Sigma的任意覆盖空间相关联，其中连通流形的基本群是Gamma.我们的orbifold不变量有一个自然和简单的几何起源的背景下，局部常数G-等变映射从G-主丛覆盖空间的Sigma的G-流形M。我们计算了任意曲面群（可定向或不可定向，紧或非紧）的对称乘积的轨道的轨道欧拉示性数的生成函数，包括指数形式和无穷乘积形式.几何上，这个无穷乘积的每个因子对应于流形Sigma的连通覆盖空间的同构类。计算的基本成分是一个结构定理的中心化同态圈积描述的自同构群的伽玛等变G-主丛在有限的伽玛集。作为推论，我们得到了组合群论中的许多恒等式。作为一个副产品，我们证明了一个简单的公式，它计算的共轭类的数量的子群在任何一个组的给定指数。我们的研究是基于轨道共形域论。
Classification of aut-fixed subgroups in free-abelian times surface groups
本文对曲面基本群与自由阿贝尔群的直积的自同构固定子群给出完全分类，并揭示其与双曲群的差异，进而应用于构造无Jiang性质的流形。
Central extensions and proper actions on products of hyperbolic spaces
本文利用拟态射研究群的中心扩张与群作用，确定欧拉类的有界性是中心扩张下保持QT性质的精确障碍，还介绍了PH性质并给出相关结果及例子。
Spherical $p$-group complexes arising from finite groups of Lie type
本文研究了有限群的p-群复形与李型群的同调关系，证明了在特定条件下，这些复形与球面建筑相关联，并展示了欧拉特征非零的结论。
Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ has a convergent Laurent expansion at $n=\infty$ involving maps on surfaces and $L^{2}$-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning $\mathrm{U}\left(n\right)$, yet with some surprising …
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