Solvable Groups of Exponential Growth and HNN Extensions
Roger Alperin
我们证明了以下版本的米尔诺定理的可解群的指数增长性：一个非循环生成的可解群包含一个上升的HNN扩展。因此，一个ERF生成的可解群是ERF当且仅当它是多环的。
We prove the following version of Milnor's theorem on solvable groups of exponential growth: A finitely generated solvable group which is not polycyclic contains an ascending HNN extension. Consequently, a finitely generated solvable group is ERF iff it is polycyclic.
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Non proper HNN extensions and uniform uniform exponential growth
如果一个无挠的群K具有K的所有无挠群S都是小的或具有一致有界的增长常数的性质，那么K的一个非真HNN扩张G，即K与整数的半直积，也具有同样的性质.这里小意味着循环的，或者，如果自同构没有周期共轭类，有界秩的自由阿贝尔。
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.
Non-linear, solvable, residually $p$ groups
本文构造了有限生成非线性可解剩余2群的第一个例子，填补了从剩余有限群到剩余p群这一研究方向的空白。
We study ascending HNN-extensions $G$ of finitely generated free abelian groups: examples of such $G$ include soluble Baumslag-Solitar groups and fundamental groups of orientable prime $3$-manifolds modelled on Sol geometry. In particular, we study the elliptic subgroup $A \leq G$, consisting of all elements that stabilise a point in the Bass-Serre tree of $G$. We consider the density of $A$ with respect to ball counting measures corresponding to finite generating sets of $G$, and we show that $A$ is exponentially negligible in $G$ with respect to such sequences of measures. As a consequence, …
Relative hyperbolicity of ascending HNN extension of groups
该研究证明了带自由因子系统的有限生成群的保系统单自同态对应的升序HNN扩张的相对双曲性，并得出有限秩自由群指数增长单自同态的升序HNN扩张相对双曲的推论，推进了几何群论相关研究。
Exponent equations in HNN-extensions
本文研究有限生成群中的指数方程，针对稳定字母作用平凡的HNN-扩张，证明在额外技术条件下背包半线性性从基群转移至扩张群，覆盖中心化子、双曲群拟凸子群等场景。
Some solvable automaton groups
它表明，有限秩的自由阿尔贝群的某些上升HNN扩展，以及各种点灯群，可以实现为自动机群，即，可以被赋予自相似结构。这包括可解的Baumslag-Solitar群BS（1，m），其中m不是+/-1。此外，还证明了对任意互素整数m，n ≥ 2，Baumslag-Solitar群BS（1，m）和BS（1，n）对可由一对对偶自动机实现.然后，使用的例子来说明施赖尔图，自动机和对偶自动机的组合之间的更一般的连接。
Elementary amenable groups of cohomological dimension 3
研究显示，余维数为3的无 torsion 初等可除群是可解群，导出长度不超过3。此类群包括所有余维数为3的可解群，且后者子类群要么是多项式群，要么是半直接积或恰当递增的 HNN 扩张。
On uniform exponential growth for solvable groups
利用J.Groves的一个定理，给出了可解群的Osin一致指数增长性的乒乓证明.我们讨论缓慢的指数增长性，并表明，这种现象消失，通过一个有限的指数子群。
Minimal exponential growth rates of metabelian Baumslag-Solitar groups and lamplighter groups
证明了对任意素数$p\geq 3$，Baumslag-Solitar群$BS(1,p)$和点灯群$\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$的最小指数增长性相等.我们还表明，$p=2$这种说法是不正确的，$BS(1,2)$的增长率等于正根的$x^3-x^2-2$，而一个的点灯组$\mathcal{L}_2$是等于黄金分割率$(1+\sqrt5)/2$。后一个值还用于表明，来自[Mann，Journal of Algebra 326，no. 1（2011）208- 217]的A.Mann的用于非半直接HNN扩展的增长率的下界是最优的。
We prove a normal form result for the groupoid of germs generated by PSL(2,C) and the exponential map. As consequences, we generalize a result of Cohen about the group of translations and powers, and prove that the subgroup of Homeo(R,+infinity) generated by the positive affine maps and the exponential map is isomorphic to a HNN-extension.
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…
On conjugacy growth for solvable groups
证明了非虚幂零的可解群具有指数共轭增长。
Exponential and weakly exponential subgroups of finite groups
本文研究有限群的指数子群与弱指数子群，分类了exp-单群，证明所有可解群是wexp-可解的，并完全分类了wexp-可解的PSL(2,q)群，给出素数密度极限为1/4。
Algebraic hulls of solvable groups and exponential iterated integrals on solvmanifolds
利用不变微分形式的Miller指数迭代积分（Chen迭代积分的推广）表示解流形$G/Γ$的代数壳的坐标环（它是幂零群的Malcev完备化对可解群的推广）.
Let $G$ be a connected exponential Lie group and $R$ be the solvable radical of $G$. We describe a condition on $G/R$ under which one can then conclude that $R$ is an exponential Lie group. The condition holds in particular when $G$ is a complex Lie group and this yields a stronger version of a result of Moskowitz and Sacksteder \cite{MS} on the center of a complex exponential Lie group being connected. Along the way we prove a criterion for elements from certain subsets of a solvable Lie group to be exponential, which would be of independent interest.
Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in P for any group of Fitting len…
Uniform Exponential Growth for Linear Groups
本文证明了特征为零的域上的任意非虚可解线性群具有一致指数增长性。
Groups acting on horocyclic products
研究群在horocyclic积上的几何作用，发现其结构取决于该积空间视觉边界的连通性，可能为有限生成虚-nilpotent群的上升HNN扩张或非有限呈现群。
Rigidity of lattices and syndetic hulls in solvable Lie groups
首先设$G$是一个完全可解的李群。我们回顾了以下结果的证明：$G$的任何闭子群在$G$中具有唯一的联会壳。由此我们得出结论：$G$的任何一致子群$Γ$在G意义下是强刚性的. D.莫斯托：如果$α:Γ\to G$是李群的同态使得$α(Γ)$在$G$中是一致的，则存在$G$的自同构$φ$使得$φ\,|\,Γ=α$。现在让$G$是一个任意（指数）可解的李群。讨论了$G$的闭子群存在联合壳的充分条件。
Invariant differential operators and central Fourier multipliers on exponential Lie groups
设$G$是指数可解李群。根据定义，$G$是$\ast$-正则的，如果对于$G$的所有酉表示$π$，$ker_{L^1(G)}π$在$ker_{C^\ast(G)}π$中是稠密的。Boidol用纯代数条件刻画了$\ast$-正则指数李群.在本文中，我们将重点关注非$\ast$-正则群。我们说$G$是本原$\ast$-正则的，如果对所有的不可约表示都满足上述密度条件.我们的目标是开发适当的工具来验证这种较弱的属性。为此，我们将介绍Duflo对$(W,p)$和中央傅立叶乘法器$ψ$上的稳定器$M=G_fN$的代表$π=\mcK(f)$在一般的位置。利用Littlewood-Paley理论，我们将得到一些关于乘子算子$T_ψ$的结果，这些结果可能是独立有趣的。通过详细研究两个重要的例子，我们将勾勒出分离三元组$(W,p,ψ)$的方法的范围。值得注意的是，"分离三元组"和"对子元的限制"方法足以证明所有维数为$\le 7$的指数可解李群都是本原$\ast$-正则的.
Let $G$ be an acylindrically hyperbolic group and $E$ an exponential equation over $G$. We show that if $E$ is solvable in $G$, then there exists a solution whose components, corresponding to loxodromic elements, can be linearly estimated in terms of lengths of the coefficients of $E$. We give a more precise answer in the case where $G$ is a relatively hyperbolic group. Under some assumption of general character, the solvability and the search problems for exponential equations over $G$ can be reduced to the peripheral subgroups of $G$.
Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $Γ\leq \textup{Mod}(Σ)$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $ρ$ of $\textup{Mod}(Σ)$ such that $ρ(Γ)$ is not virtually solvable. We then apply results of Lubotzky and Meiri to show that for any random walk on such a group the probability of landing on a power, or on an element with topological entropy $0$ both decrease exponentially in the length of the walk.
The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (2002) where they showed that this problem is in polynomial time for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups $G$ having a nilpotent normal subgroup $H$ with nilpotent factor $G/H$. This paper shows that such normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless…
Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the conjugation action is $\mathbb{F}$-linear. For $k\in \mathbb N$ we describe a necessary and sufficient condition for all elements from any coset $xN$, $x\in G$, to admit $k$-th roots in $G$, in terms of the action of $x$ on the quotients $N_j/N_{j+1}.$ This yields in particular a condition for surjectivity of the power maps, generalising various results known in sp…
On conjugacy growth of linear groups
本文研究了非线性生成的线性群的共轭增长性。我们证明了GL_d的非虚可解子群具有一致的指数共轭增长，并且实际上，对于字度量，作为半径为n的球的元素的特征多项式而产生的不同多项式的数目仅以维数d有界地从0以指数增长性增长.
Genericity of Filling Elements
若F（X）在$\mathbb{R}$ -树上的每一个极小作用中都有正的平移长度，则称F（X）中的一个元素是满的.本文证明了F（X）的填充元集在Arzhantseva和Ol'shanskii意义下是指数F（X）-泛的.我们还提供了一个代数充分条件的元素被填充，并表明存在一个指数F（X）-通用的填充元素的子集，其成员问题是在线性时间可解的。
Sieve methods in group theory I: Powers in Linear groups
给出了群的一般筛法。它使人们能够“测量”一个有限生成群的子集。作为应用，我们证明了如果$Γ$是一个特征为零的非虚可解线性群，则$Γ$的真幂集是指数小的.这是对\cite{HKLS}的主要结果的深远加强。
On real and rational elements in a class of Lie groups
本文为域上包括李群、代数群和有限群在内的一类群提供了确定实元素与有理元素的统一方法，并应用于半直积SL(2,R)⋉Sym^n(R^2)的分类，同时研究了连通实可解李群中实元素与单连通性、可解性及指数性的关系。
On the dimension growth of groups
Gromov在1999年引入了群的维数增长函数。我们证明了R的每个可解的n-生成子群。Thompson群$F$具有多项式维数增长，而群$F$本身和某些第3类可解群具有指数维数增长，并受指数控制.我们描述了有限图的维数增长、扩张性质和拉姆齐理论之间的联系。
We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many fat bigons", and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the fo…
Random subgroups of linear groups are free
我们证明了在一个任意的非虚可解线性群上，任意两个独立的随机游动最终生成一个自由子群。事实上，这将适用于指数数量的独立随机游动。
We give a solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups of exponential volume growth. For smaller classes, we give solutions among solvable groups. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula …
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