Khalid Bou-Rabee, Samuel Young
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.
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We prove that for any natural n>1, the abstract commensurator group of the Baumslag - Solitar group BS(1,n) is isomorphic to the group of 2 by 2 upper triangular matrices A over rational numbers with A_{11}=1. We also prove that for any finitely generated group G with the unique root property the natural homomorphisms Aut(G)--> Comm(G)--> QI(G) are embeddings.
A survey of results on the residual properties of Baumslag -- Solitar groups which have been obtained to date.
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。
The arguments of Cannon, Floyd, Grayson and Thurston showing that solve geometry groups are not almost convex also show that solvable Baumslag-Solitar groups are not almost convex.
We compute estimates for the word metric of Baumslag--Solitar groups in terms of the Britton's lemma normal form. As a corollary, we find lower bounds for the growth rate for the groups $BS(p,q)$, with $1<p\le q$.
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.
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.
We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different Baumslag-Solitar groups and is not multiplicative. This allows us to give an example of a finitely generated profinite group which is not virtually pronilpotent but whose zeta function has an Euler product.
The Baumslag-Solitar groups and their certain variations are a-T-menable. This is proved by embeding them into topological groups and studying representation theoretic properties of the latter. The paper is motivated by the questions of A. Valette.
We give a language of unique geodesic normal forms for the Baumslag-Solitar group BS(1,2) that is context-free and 1-counter. We discuss the classes of context-free, 1-counter and counter languages, and explain how they are inter-related.
We prove that groups in a certain class of metabelian locally compact groups, have quadratic Dehn function. As an application, we embed the solvable Baumslag-Solitar groups into finitely presented metabelian groups with quadratic Dehn function. Also, we prove that Baumslag's finitely presented metabelian groups, in which the lamplighter groups embed, have quadratic Dehn function.
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.
In this paper we give necessary conditions on group presentations, with two generators and one relator, in order to be the group of a virtual knot diagram. Although those conditions are not enough, we use them to determine, completely, whether or not a Baumslag-Solitar group is the group of a $2$-bridge virtual knot. Moreover, we present a combinatorial proof of the fact that these groups are not $2$-bridge classical knot groups.
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。
Our main result is that many triangles of Baumslag-Solitar groups collapse to finite groups, generalizing a famous example of Hirsch. A triangle of Baumslag-Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of class<4. We…
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。
We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that BS(m,n) --> F_2 for |m|,|n| --> \infty and BS(1,n) --> Z \wr Z for |n| --> \infty. For m fixed, |m|>1, we show that the sequence (BS(m,n))_n is not convergent and characterize many convergent subsequences. Moreover if X_m is the set of BS(m,n)'s for n relatively prime to m and |n|>1, then the map BS(m,n) \mapsto n extends continuously on the closure of X_m to a surjection onto invertible m-adic integers.
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。
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…
This paper studies automatic structures for subsemigroups of Baumslag--Solitar semigroups (that is, semigroups presented by $\ < x,y \mid (yx^m, x^ny)\ >$, where $m$ and $n$ are natural numbers). A geometric argument (a rarity in the field of automatic semigroups) is used to show that if $m \gt n$, all of the finitely generated subsemigroups of this semigroup are [right-] automatic. If $m \lt n$, all of its finitely generated subsemigroups are left-automatic. If $m = n$, there exist finitely generated subsemigroups that are not automatic. An appendix discusses the implications of these results…
We construct an invariant deformation retract of a deformation space of G-trees. We show that this complex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this complex we compute the automorphism group of the classical non-solvable Baumslag-Solitar groups BS(p,q). The most interesting case is when p properly divides q. Collins and Levin computed a presentation for Aut(BS(p,q)) in this case using algebraic methods. Our computation uses Bass-Serre theory to derive these presentations. Additionally, we provide a geometric argument showing Out(BS…
For each Baumslag-Solitar group BS(m,n) (m,n nonzero integers), a totally disconnected, locally compact group, G_{m,n}, is constructed so that BS(m,n) is identified with a dense subgroup of G_{m,n}. The scale function on G_{m,n}, a structural invariant for the topological group, is seen to distinguish the parameters m and n to the extent that the set of scale values is {(lcm(m,n)/|m|)^ρ, (lcm(m,n)/|n|)^ρ | ρ\in N}. It is also shown that G_{m,n} has flat rank 1 when |m|\neq |n| and 0 otherwise, and that G_{m,n} has a compact, open subgroup isomorphic to the product {(Z_p,+) | p is a prime divis…
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。
We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidire…
Generalized Baumslag-Solitar groups (GBS groups) are groups that act on trees with infinite cyclic edge and vertex stabilizers. Such an action is described by a labeled graph (essentially, the quotient graph of groups). This paper addresses the problem of determining whether two given labeled graphs define isomorphic groups; this is the isomorphism problem for GBS groups. There are two main results and some applications. First, we find necessary and sufficient conditions for a GBS group to be represented by only finitely many reduced labeled graphs. These conditions can be checked effectively …
A generalized Baumslag-Solitar group is the fundamental group of a graph of groups all of whose vertex and edge groups are infinite cyclic. Levitt proves that any generalized Baumslag-Solitar group has property R-infinity, that is, any automorphism has an infinite number of twisted conjugacy classes. We show that any group quasi-isometric to a generalized Baumslag-Solitar group also has property R-infinity. This extends work of the authors proving that any group quasi-isometric to a solvable Baumslag-Solitar BS(1,n) group has property R-infinity, and relies on the classification of generalized…
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。
To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided.
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。
Let $Γ$ be the fundamental group of a surface of finite type and Comm$(Γ)$ be its abstract commensurator. Then Comm$(Γ)$ contains the solvable Baumslag--Solitar groups $\langle a ,b : a b a^{-1} = b^n \rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ has an image in Comm$(Γ)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(Γ)$ are concrete objects amenable to computational methods. For example, we give a proof that $\langle a ,b : a b^2 a^{-1} = b^3 …
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 study qualitative properties of the group von Neumann algebra of a Baumslag-Solitar group. Namely, we prove that, in the non-amenable and {ICC} case, the associated ${\rm II}_1$ factor is prime, not solid, and does not have any Cartan subalgebra.
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…
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。
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。
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 study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (`fully reduced' decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag-Solitar groups with no non-trivial integral moduli.
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.
We classify the finite-dimensional irreducible linear representations of the Baumslag-Solitar groups BS(p,q) = < a, b | a b^p = b^q a > for relatively prime p and q. The general strategy of the argument is to consider the matrix group given by image of a representation and study its Zariski closure in GL(n, C).
We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).
We prove that the power word problem for certain metabelian subgroups of $\mathsf{GL}(2,\mathbb{C})$ (including the solvable Baumslag-Solitar groups $\mathsf{BS}(1,q) = \langle a,t \mid t a t^{-1} = a^q \rangle$) belongs to the circuit complexity class $\mathsf{TC}^0$. In the power word problem, the input consists of group elements $g_1, \ldots, g_d$ and binary encoded integers $n_1, \ldots, n_d$ and it is asked whether $g_1^{n_1} \cdots g_d^{n_d} = 1$ holds. Moreover, we prove that the knapsack problem for $\mathsf{BS}(1,q)$ is $\mathsf{NP}$-complete. In the knapsack problem, the input consis…
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.
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