Baumslag Solitar Groups BS Research Articles

Using edt0l systems to solve some equations in the solvable Baumslag-Solitar groups

We investigate the solution sets to equations in the solvable Baumslag-Solitar groups BS(1,k), k≥2, 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.

Shortcut graphs and groups

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional CAT(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions.

On the domino problem of the Baumslag-Solitar groups

In [1] we construct aperiodic tile sets on the Baumslag-Solitar groups BS(m,n). Aperiodicity plays a central role in the undecidability of the classical domino problem on Z2, and analogously to this we state as a corollary of the main construction that the Domino problem is undecidable on all Baumslag-Solitar groups. In the present work we elaborate on the claim and provide a full proof of this fact. We also provide details of another result reported in [1]: there are tiles that tile the Baumslag-Solitar group BS(m,n) but none of the valid tilings is recursive. The proofs are based on simulating piecewise affine functions by tiles on BS(m,n).

Random walks on Baumslag–Solitar groups

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.

Decidability of knapsack problem for Baumslag-Solitar group

In this work we investigate a decidability problem of group version of the knapsack problem for Baumslag-Solitar group BS(p, q). We proved, that the knapsack problem is decidable for group BS(p, q) for coprime integers p > 1, q > 1. In the case when p = 1, q ∈ ℕ, we proved that knapsack problem is decidable for group BS(1, q) with some restriction on the input of the problem. However, the problem of the decidability of the knapsack problem for group BS(1, q) on the whole set of inputs remains open.

K-THEORY FOR THE C*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

AbstractWe provide a new computation of the K-theory of the group C*-algebra of the solvable Baumslag–Solitar group BS(1, n) (n ≠ 1); our computation is based on the Pimsner–Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product ℤ[1/n] ⋊ ℤ. We deduce from it a new proof of the Baum–Connes conjecture with trivial coefficients for BS(1, n).

Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

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[9] for the growth rates of non-semidirect HNN extensions is optimal.

Representation and character varieties of the Baumslag-Solitar groups

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety Rn(BS(p, q)) are rational varieties of dimension n2, and each irreducible component of the character variety Xn(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety Rns (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety Rns (BS(p, q)) are isomorphic to A1 {0}.

A small doubling structure theorem in a Baumslag–Solitar group

We solve a general inverse problem of small doubling type in a monoid, which is a subset of the Baumslag–Solitar group BS(1,2).

Quotients and subgroups of Baumslag–Solitar groups

Abstract 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).

From automatic structures to automatic groups

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these groups are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the word problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2-nilpotent groups and Baumslag–Solitar groups BS(1,n) are graph automatic, as well as many other metabelian groups.

The cogrowth series forBS(N, N) is D-finite

We compute the cogrowth series for Baumslag–Solitar groups BS (N, N) = 〈a,b|aNb = baN〉, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.

Embedding of baumslag-solitar groups into the generalized Baumslag-Solitar groups

A finitely generated group G that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag-Solitar group or GBS-group. Let p and q be coprime integers other than 0, 1, and −1. We prove that the Baumslag-Solitar group BS(p, q) embeds into G if and only if the equation x−1ypx = yq is solvable in G for y ≠ 1; i.e., \(\tfrac{p} {q} \) ∈ Δ(G), where Δ is the modular homomorphism.

Irreducible representations of subgroups of finite index in Baumslag-Solitar groups

We describe all finite-dimensional irreducible linear representations over the field of complex numbers for an arbitrary subgroup of finite index of a Baumslag-Solitar group BS(p, q) = 〈a, t | t−1apt = aq〉 for coprime p and q. We find necessary and sufficient conditions for the equivalence of these representations, using only standard facts of linear algebra and the description of subgroups of finite index in Baumslag-Solitar groups.

Efficient Algorithms for Highly Compressed Data: The Word Problem in Generalized Higman Groups Is in P

This paper continues the 2012 STACS contribution by Diekert, Ushakov, and the author as well as the 2012 IJAC publication by the same authors. We extend the results published there in two ways. First, we show that the data structure of power circuits can be generalized to work with arbitrary bases q?2. This results in a data structure that can hold huge integers, arising by iteratively forming powers of q. We show that the properties of power circuits known for q=2 translate to the general case. This generalization is non-trivial and additional techniques are required to preserve the time bounds of arithmetic operations that were shown for the case q=2. The extended power circuit model permits us to conduct operations in the Baumslag-Solitar group BS(1,q) as efficiently as in BS(1,2). This allows us to solve the word problem in the generalization H 4(1,q) of Higman's group in polynomial time. The group H 4(1,q) is an amalgamated product of four copies of the Baumslag-Solitar group BS(1,q) rather than BS(1,2) in the original group H 4=H 4(1,2). As a second result, we allow arbitrary numbers f?4 of copies of BS(1,q), leading to an even more generalized notion of Higman groups H f (1,q). We prove that the word problem of the latter can still be solved within the ${\mathcal {O}}(n^{6})$ time bound that was shown for H 4(1,2).

The number of finite index subgroups of Baumslag–Solitar groups

Gelman obtained a simple formula for the number of finite index subgroups of Baumslag–Solitar groups BS(p, q) = 〈a, t | t −1 a p t = a q 〉, where p and q are coprime integers. We generalize this formula to the case of arbitrary nonzero integers. The proof is obtained by calculating the number of permutations y ∈ S n such that the subgroup of S n generated by x and y, where x is given, is transitive. Bibliography: 4 titles.

Two remarks on the first-order theories of Baumslag-Solitar groups

We characterize all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS(m, 1). It turns out that a finitely generated group G is elementarily equivalent to BS(m, 1) if and only if G is isomorphic to BS(m, 1). 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.

Rigidity of certain solvable actions on the sphere

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure.

Abstract Commensurators of Solvable Baumslag–Solitar Groups

We prove that for any n ≥ 2, the abstract commensurator group of the Baumslag–Solitar group BS(1, n) is isomorphic to the subgroup of GL2(ℚ). 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.

Growth in Baumslag–Solitar groups I: subgroups and rationality

AbstractThe computation of growth series for the higher Baumslag–Solitar groups is an open problem first posed by de la Harpe and Grigorchuk. We study the growth of the horocyclic subgroup as the key to the overall growth of these Baumslag–Solitar groups BS(p,q), where 1<p<q. In fact, the overall growth series can be represented as a modified convolution product with one of the factors being based on the series for the horocyclic subgroup. We exhibit two distinct algorithms that compute the growth of the horocyclic subgroup and discuss the time and space complexity of these algorithms. We show that when p divides q, the horocyclic subgroup has a geodesic combing whose words form a context-free (in fact, one-counter) language. A theorem of Chomsky–Schützenberger allows us to compute the growth series for this subgroup, which is rational. When p does not divide q, we show that no geodesic combing for the horocyclic subgroup forms a context-free language, although there is a context-sensitive geodesic combing. We exhibit a specific linearly bounded Turing machine that accepts this language (with quadratic time complexity) in the case of BS(2,3) and outline the Turing machine construction in the general case.