A linear-time algorithm to compute geodesics in solvable Baumslag-Solitar groups
Murray Elder
我们提出了一个算法来转换一个字的长度$n$在标准生成器的可解Baumslag-Solitar组$BS(1,p)$到测地线字，它运行在线性时间和$O(n\log n)$空间上的随机接入机。
We present an algorithm to convert a word of length $n$ in the standard generators of the solvable Baumslag-Solitar group $BS(1,p)$ into a geodesic word, which runs in linear time and $O(n\log n)$ space on a random access machine.
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On some residual properties of Baumslag -- Solitar groups
迄今为止已获得的Baumslag-Solitar群的剩余性能结果的调查。
In this paper we classify Baumslag-Solitar groups up to commensurability. In order to prove our main result we give a solution to the isomorphism problem for a subclass of Generalised Baumslag-Solitar groups.
This paper gives asymptotic formulas for the subgroup growth and maximal subgroup growth of all Baumslag-Solitar groups.
Relative profinite rigidity of Baumslag-Solitar groups
本文研究Baumslag-Solitar群的相对有限完备刚性，证明剩余有限情形可仅凭有限商在同类群中被唯一识别。
Solvable Baumslag-Solitar Groups Are Not Almost Convex
Cannon、Floyd、Grayson和Thurston的论证表明，可解几何群不是几乎凸的，也表明可解Baumslag-Solitar群不是几乎凸的。
Any non-residually finite Baumslag-Solitar group has a non-residually finite image in the abstract commensuration of a nonabelian free group. This gives a new proof (avoiding Britton's Lemma) of the classification of residually finite Baumslag-Solitar groups.
Metric properties of Baumslag-Solitar groups
我们计算的字度量的Baumslag-Solitar集团的Britton引理范式的估计。作为一个推论，我们发现的增长率为组$BS(p,q)$，与$1<p\le q$的下界。
We find the lower central series for residually nilpotent Baumslag-Solitar groups, and find the intersection of all terms of the lower central series. Also, we find non-abelian Bauslag-Solitar groups for which the lower central series has length 2. For some Baumslag-Solitar groups a connection is found between the intersection of all subgroups of finite index and the intersection of all terms of the lower central series.
A formula for the normal subgroup growth of Baumslag-Solitar groups
当p和q互质时，给出了Baumslag-Solitar群BS（p，q）中每个有限指标的正规子群个数的精确公式.与所有有限指数子群的公式不同，这个公式区分了不同的Baumslag-Solitar群，并且不是乘法的。这使我们能够给出一个例子，一个非线性生成的投射有限群，它不是虚幂零的，但它的zeta函数有一个欧拉积。
a-T-menability of Baumslag-Solitar groups
Baumslag-Solitar群和它们的某些变体是可解释的。通过将它们嵌入到拓扑群中并研究拓扑群的表示论性质，证明了这一点。本文是由A.瓦莱特
A context-free and a 1-counter geodesic language for a Baumslag-Solitar group
本文给出了Baumslag-Solitar群BS（1，2）的唯一测地正规形的一种语言，它是上下文无关的和1-计数器的.我们讨论上下文无关，1-计数器和计数器语言的类，并解释它们是如何相互关联的。
Abstract commensurators of solvable Baumslag - Solitar groups
证明了对任意自然数n>1，Baumslag-Solitar群BS（1，n）的抽象分解子群同构于A_{11} =1的有理数上的2 × 2上三角形矩阵A.证明了对任何具有唯一根性质的有限生成群G，其自然同态Aut（G）--> Comm（G）--> QI（G）是嵌入.
Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups
本文证明了亚阿贝尔局部紧群中的群具有二次Dehn函数。作为应用，我们将可解Baumslag-Solitar群嵌入到具有二次Dehn函数的可表示亚交换群中。此外，我们还证明了灯群嵌入其中的Baumslag有限表现亚阿贝尔群具有二次Dehn函数。
Similarly to the action of $Out(F_N)$ on Outer Space, the outer automorphism group of a Generalized Baumslag Solitar group acts on a deformation space endowed with the Lipschitz metric and the action of any fully irreducible automorphism with a train track representative is hyperbolic. Inspired by previous work by Algom-Kfir, we prove that the axis of such an automorphism is strongly contracting.
We investigate the solution sets to equations in the solvable Baumslag-Solitar groups $BS(1,k)$, $k\geq2$, and show that these sets are represented by EDT0L languages in some cases. In particular, we prove that the multiplication table of such a group forms an EDT0L language with respect to a specific natural normal form for group elements.
A group $G$ has the $R_{\infty}$--property if the number $R(φ)$ of twisted conjugacy classes is infinite for any automorphism $φ$ of $G$. For such a group $G$, the $R_{\infty}$--nilpotency index is the least integer $c$ such that $G/γ_{c+1}(G)$ still has the $R_\infty$--property. In this paper, we determine the $R_{\infty}$--nilpotency degree of all Baumslag--Solitar groups.
On the isomorphism problem for generalized Baumslag-Solitar groups: invariants and flexible configurations
本研究针对具有一个拟共轭类且完全支撑带隙的广义Baumslag-Solitar（GBS）群，引入一族不变量，证明其同构问题可判定，且该不变量可完全刻画此类群的同构性。
The intersections of the lower central series and the subgroups of finite p-index of Generalized Baumslag-Solitar tree groups
本文针对底图为树的广义Baumslag-Solitar群，计算了其下中心序列的交γ_ω(G)及有限p-指数子群的交(N_p)_ω(G)。
For a Baumslag-Solitar group $G$ we calculate the intersection $γ_w(G)$ of all terms of the lower central sequence of $G$.Using this we are able to show that $[γ_w(G),G]=γ_w(G)$ thus answering a question of Bardakov and Neschadim. Finally we show that the quotient groups $γ_c(G)/γ_{c+1}(G)$ of the lower central series of $G$ are finite.
We compute the Bieri-Neumann-Strebel invariants $Σ^1$ for the generalized solvable Baumslag-Solitar groups $Γ_n$ and their finite index subgroups. Using $Σ^1$, we show that certain finite index subgroups of $Γ_n$ cannot be isomorphic to $Γ_{k}$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_\infty$ for the groups $Γ_n$ and their finite index subgroups.
We define fully irreducible automorphisms of generalized Baumslag-Solitar groups in analogy with fully irreducible automorphisms of free groups. We first obtain a characterization of fully irreducible automorphisms analogous to a condition given by Kapovich. Next we discuss the existence of pseudo-periodic conjugacy classes based on the study of Nielsen paths in a train track representative. Finally we give an algorithm which finds out whether an automorphism with a train track representative is pseudo-atoroidal and fully irreducible.
Some Baumslag-Solitar groups are two bridges virtual knots
本文给出了具有两个生成元和一个关系元的群表示成为虚纽结图群的必要条件。虽然这些条件是不够的，我们使用它们来确定，完全，是否Baumslag-Solitar群是一个$2$桥虚拟结的组。此外，我们提出了一个组合证明的事实，这些群体是不是$2$ -桥经典纽结群。
The isomorphism problem for random generalized Baumslag-Solitar groups with many edges
研究广义鲍姆slag - 索利塔群同构问题，通过证明随机模型中具多条边的GBS图大概率灵活，利用其同构问题可判定性，得出随机GBS群高概率可判定的结论。
Generalised Baumslag-Solitar groups and Hierarchically Hyperbolic Groups
本文研究任意维数广义Baumslag-Solitar群在双曲空间上的等距作用，证明实质上成为分层双曲群及性质（QT）在拟等距下均非不变量。
Gromov's Conjecture for Product of Baumslag-Solitar groups and some other One-relator groups
证明Baumslag-Solitar群及部分推广不可定向曲面群的单关系群是虚拟2-可避免的，结合前期结果，证实了基本群为这些群乘积的闭定向自旋流形的Gromov PSC流形万有覆盖宏观维数猜想。
Triangles of Baumslag-Solitar Groups
我们的主要结果是Baumslag-Solitar群的许多三角形坍缩为有限群，推广了Hirsch的一个著名例子。Baumslag-Solitar群的三角形表示一个有三个生成元的群，循环有序，每个生成元将前一个生成元的某个幂共轭到另一个幂。有六个参数，成对发生，我们表明，三角形未能发展时，其中一个参数划分其合作伙伴，除了少数特殊情况。此外，在相当一般的条件下，这个群是有限的，并且是<4类的可解群。即使在我们不能确定可展性的情况下，我们也得到了许多关于有限导数的信息。
Self-similarity of the generalized Baumslag-Solitar groups
本文证明了所有有限余有限的广义巴乌斯-索尔itar群均为自相似群，并进一步证明了某些图的fundamental群也是自相似群。
In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $k\geq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulas for the series. As a result of our computation we also establish that in each $BS(1,k)$ the conjugacy and standard growth rates are equal.
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.
On the isomorphism problem for generalized Baumslag-Solitar groups: angles
本文为广义Baumslag-Solitar群引入新的同构不变量“极限角”，具有动力学解释，并用于分类一个顶点两条边的GBS群。
Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups
该研究证明了Baumslag-Solitar群的极限的多项代数性质，给出其表现、同构分类、自同构描述及相关判定问题讨论，并证明其在双生成元带标群空间中具有非零Hausdorff维数。
Convergence of Baumslag-Solitar groups
在标记群空间中研究了Baumslag-Solitar群的收敛序列.本文证明了BS（m，n）--> F_2对于|m|，|n| --> \infty和BS（1，n）--> Z \wr Z对于|n| --> \infty .当m固定，|m| >1时，我们证明了序列（BS（m，n））n不收敛，并刻画了许多收敛序列.此外，若X_m是BS（m，n）的集合，其中n与m互素，且|n| >1，则映射BS（m，n）\mapsto n在X_m的闭包上连续扩张到可逆m-adic整数上的满射.
We study the HNN extension of $\mathbb{Z}^m$ given by the cubing endomorphism $g\mapsto g^3$, and prove that such groups have rational growth. To do so, we describe a method of computing the subgroup growth series of the horocyclic subgroup $\mathbb{Z}^m$ in this family of examples, prove that for all $m$ the group has rational growth. In the appendix, the subgroup growth series has been computed for all $m \leq 10$.
Limits of Baumslag-Solitar groups
本研究对带典范生成元集的Baumslag-Solitar群的极限给出m进整数参数化，证明其在可逆m进整数上连续单射，明确这类极限的群扩张结构与有限展示性，并给出相关树上自然作用的展示。
Cohomological Separability of Baumslag--Solitar groups and Their Generalisations
本文在广义 Baumslag–Solitar 群类中给出上同调可分性的等价刻画，并对 Baumslag–Solitar 群得到上同调可分性与有限完备化上同调维数的三分法，从而构造出具有和不具有可分上同调的非剩余有限单关系群例子。
A note on twisted conjugacy and generalized Baumslag-Solitar groups
广义Baumslag-Solitar群是群的图的基本群，其顶点群和棱道群都是无限循环的。Levitt证明了任何广义Baumslag-Solitar群都具有R-无穷性质，即任何自同构都有无穷多个扭共轭类。证明了任何拟等距于广义Baumslag-Solitar群的群也具有R-无穷性质。这扩展了作者的工作，证明了任何拟等距于可解Baumslag-Solitar BS（1，n）群的群都具有R-无穷性质，并依赖于Whyte给出的广义Baumslag-Solitar群的分类。
Invariants of orbit equivalence relations and Baumslag-Solitar groups
对于标准概率空间上的非顺从Baumslag-Solitar群的遍历的、本质自由的和保测度作用，流是相关联的.流的同构类是Baumslag-Solitar群在弱轨道等价下的不变量。还提供了测量等效于Baumslag-Solitar群的群的结果。
We say a group $G$ has property $R_\infty$ if the number $R(φ)$ of twisted conjugacy classes is infinite for every automorphism $φ$ of $G$. For such groups, the $R_\infty$-nilpotency degree is the least integer $c$ such that $G/γ_{c+1}(G)$ has property $R_\infty$. In this work, we compute the $R_\infty$-nilpotency degree of all Generalized Solvable Baumslag-Solitar groups $Γ_n$. Moreover, we compute the lower central series of $Γ_n$, write the nilpotent quotients $Γ_{n,c}=Γ_n/γ_{c+1}(Γ_n)$ as semidirect products of finitely generated abelian groups and classify which integer invertible matrice…
Automatic structures for subsemigroups of Baumslag--Solitar semigroups
本文研究Baumslag-Solitar半群（即由$\ < x,y \mid (yx^m, x^ny)\ >$提出的半群，其中$m$和$n$是自然数）的子半群的自动结构.利用一个几何论证（在自动半群领域中很少见）证明了：如果$m \gt n$，则这个半群的所有逆生成子半群都是[右-]自动的.若$m \lt n$，则它的所有逆生成子半群都是左自动的。若$m = n$，则存在非自动生成的子半群.附录讨论了这些结果的影响Malcev演示文稿的理论。（马尔切夫表示是可嵌入群的半群的一种特殊表示。
Deformation spaces of G-trees and automorphisms of Baumslag-Solitar groups
我们构造了G-树变形空间的不变形变收缩核。我们表明，这个复杂的是有限维在某些情况下，并提供了一个例子，是不是有限维的。利用这个复形，我们计算了经典不可解Baumslag-Solitar群BS（p，q）的自同构群.最有趣的情况是当p整除q时。Collins和Levin在这种情况下使用代数方法计算了Aut（BS（p，q））的表示。我们的计算使用Bass-Serre理论来推导这些演示文稿。此外，我们还提供了一个几何参数，表明当p正确整除q时，Out（BS（p，q））不是随机生成的。
Totally disconnected groups from Baumslag-Solitar groups
对于每个Baumslag-Solitar群BS（m，n）（m，n个非零整数），构造一个全不连通的局部紧群G_{m,n}，使得BS（m，n）与G_{m,n}的稠密子群相同. G_{m,n}上的标度函数（拓扑群的结构不变量）被认为在标度值的集合为{（lcm（m，n）/ |m|）^p，（lcm（m，n）/ |n|）^p的程度上区分参数m和n| ρ \in N}。当|m| \neq |n|时，G_{m,n}的秩为1，否则秩为0，并且G_{m,n}有一个同构于乘积{（Z_p，+）}的紧的开子群|p是标度的素因子}。
Fixed subgroups of generalised Baumslag-Solitar groups
本文研究广义Baumslag-Solitar群的自同构固定子群，刻画了在边稳定子严格包含于顶点稳定子条件下存在非有限生成固定子群的群，并证明有限阶自同构及树图情形下固定子群均有限生成。
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。
On the isomorphism problem for generalized Baumslag-Solitar groups
广义Baumslag-Solitar群（GBS群）是作用在具有无限循环边和顶点稳定子的树上的群。这样的动作用一个标号图（本质上是群的商图）来描述。本文讨论了确定两个给定的标号图是否定义同构群的问题;这就是GBS群的同构问题。有两个主要结果和一些应用。首先，我们找到了一个GBS群仅由1/2个约化标号图表示的充要条件。这些条件可以从任何标号图有效地检查。然后证明了当GBS群的标号图的第一个贝蒂数至多为1时，同构问题是可解的.
A note on the von Neumann algebra of a Baumslag-Solitar group
研究了Baumslag-Solitar群的群von Neumann代数的定性性质。也就是说，我们证明了，在非顺从和{ICC}的情况下，相关的${\rm II}_1$因子是素数，不是固体，没有任何Cartan子代数。
Solvable Baumslag-Solitar Lattices
本文对可解Baumslag-Solitar群模型空间$X_n$等距群中的格进行了完整分类，并证明其虽不满足强刚性，但满足一种较弱的刚性形式。
The isomorphism problem for small-rose generalized Baumslag-Solitar groups
该研究聚焦群论中的同构判定问题，证明了小玫瑰型非上升广义Baumslag-Solitar群的同构问题可解，为这类有限生成群的结构研究提供了判定依据。
Splittings of generalized Baumslag-Solitar groups
本文从广义Baumslag-Solitar群作为无限循环群的图的基本群的（通常是非唯一的）分裂的观点出发，研究了广义Baumslag-Solitar群的结构.我们发现和表征某些分解的最小复杂性（“完全减少”分解），并给出了一个简化的变形的存在性证明。我们还证明了一个有限性定理，并解决了同构问题的广义Baumslag-Solitar群没有非平凡的积分模。
On the space of subgroups of Baumslag-Solitar groups II: High transitivity
本文研究Baumslag-Solitar群子群空间的高传递性，发现有限表型下高传递性仅在表型为1时普遍，无限表型下高传递性普遍除非|m|=|n|。
Let $G$ be a generalized Baumslag-Solitar group and $\mathcal{C}$ be a class of groups containing at least one non-unit group and closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for every $y \in Y$. We give a criterion for $G$ to be residually a $\mathcal{C}$-group provided $\mathcal{C}$ consists only of periodic groups. We also prove that $G$ is residually a torsion-free $\mathcal{C}$-group if $\mathcal{C}$ contains at least one non-periodic group and is closed under tak…
We calculate the exact values of the Følner function $\mathrm{Føl}$ of the lamplighter group $\mathbb{Z}\wr\mathbb{Z}/2\mathbb{Z}$ for the standard generating set. More generally, for any finite group $D$ and $n\geq|D|$, we obtain the exact value of $\mathrm{Føl}(n)$ on the wreath product $\mathbb{Z}\wr D$, for a generating set induced by a generator on $\mathbb{Z}$ and the entire group being taken as generators for $D$. We also describe the Følner sets that give rise to it. Følner functions encode the isoperimetric properties of amenable groups and have previously been studied up to asymptoti…
In this note we characterise all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS$(1,n)$. It turns out that a finitely generated group $G$ is elementarily equivalent to BS$(1,n)$ if and only if $G$ is isomorphic to BS$(1,n)$. Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.
Irreducible Representations of Baumslag-Solitar Groups
本文对Baumslag-Solitar群BS（p，q）= < a，b的有限维不可约线性表示进行了分类|对于互素p和q，a b^p = b^q a >。一般的论证策略是考虑由一个表示的象给出的矩阵群，并研究它在GL（n，C）中的Zapriki闭包。
Endo-Twisted Conjugacy and Outer Fixed Points in Solvable Baumslag--Solitar Groups
本文研究了可解鲍姆斯拉-索拉尔群中内扭共轭问题与外固定点的关系，提出算法并探讨其相互作用。
Quotients and subgroups of Baumslag-Solitar groups
我们确定了所有广义Baumslag-Solitar群（作用在稳定子无限循环的树上的有限生成群）是给定Baumslag-Solitar群BS（m，n）的商，以及（当BS（m，n）不是Hopfian时）它们中的哪一个也允许BS（m，n）作为商.我们确定的r，s的值可以嵌入BS（r，s）到一个给定的BS（m，n），我们的特点是有限生成群嵌入到一些BS（n，n）。
Quadratic equations in metabelian Baumslag-Solitar groups
本文研究亚交换 Baumslag–Solitar 群 $\mathbf{BS}(1,n)$ 上二次方程的丢番图问题，证明 $n\neq\pm1$ 时其为 NP 完全，并刻画可定向、不可定向等子类的复杂性。
We consider random walks on non-amenable Baumslag-Solitar groups BS(p,q) and describe their Poisson-Furstenberg boundary. The latter is a probabilistic model for the long-time behaviour of the random walk. In our situation, we identify it in terms of the space of ends of the Bass-Serre tree and the real line using Kaimanovich's strip criterion.
Groups elementarily equivalent to metabelian Baumslag$-$Solitar groups and regular bi-interpretability
本文证明了巴乌斯拉-索拉特群与整数环的正则双可解释性，并描述了与该群元素等价的所有群。
Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity
本文为保测度等价关系的非周期包含引入代数不变量，借此证明了非顺从Baumslag-Solitar群直积自由pmp作用的稳定轨道等价刚性，并完成了其相对profinite完备化直积的完全分类。
Suzuki recently gave constructions of non-discrete examples of locally compact C*-simple groups and Raum showed C*-simplicity of the relative profinite completions of the Baumslag-Solitar groups by using Suzuki's results. We extend this result to some fundamental groups of graphs of groups called generalized Baumslag-Solitar groups. In this article, we focus on some sufficient condition to show that these locally compact groups are C*-simple and that KMS-weights of these reduced group C*-algebras are unique. This condition is an analogue of the Powers averaging property of discrete groups and …
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