Baumslag Solitar Groups BS Research Articles

ON THE STRUCTURE OF SOME REDUCED AMALGAMATED FREE PRODUCT C*-ALGEBRAS

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

ON COMPUTING GEODESICS IN BAUMSLAG–SOLITAR GROUPS

We introduce the peak normal form for elements of the Baumslag–Solitar groups BS(p,q). This normal form is very close to the length-lexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u-1v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the peak normal form and the length-lexicographical normal form coincide. The main result of this paper is that we can compute the peak normal form in polynomial time if p divides q. As consequence we can compute geodesic lengths in this case. In particular, this gives a partial answer to Question 1 in [4] For arbitrary p and q it is possible to compute the peak normal form (length-lexicographical normal form resp.) also the for elements in the horocyclic subgroup and, more generally, for elements which we call hills. This approach leads to a linear time reduction of the problem of computing geodesics to the problem of computing geodesics for Britton-reduced words where the t-sequence starts with t-1 and ends with t.

Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson’s group F and the Baumslag–Solitar groups BS(1, p) with p ≥ 3. In order to make this construction for Thompson’s group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups Bm(𝕊2\{x1,…,xn}) of the n-punctured sphere includes the usual Artin braid groups Bm(for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type [Formula: see text] (for n = 3).We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of Bmis completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients.When n ≥ 2, we prove that the lower central series (respectively, derived series) of Bm(𝕊2\{x1,…,xn}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B2(𝕊2\{x1,x2}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B2(𝕊2\{x1,x2}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ2* ℤ. Further, its lower central series quotients Γi/Γi + 1are direct sums of copies of ℤ2, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γi/Γi + 1are 2-elementary finitely-generated groups.

Deformation spaces of $G$-trees and automorphisms of Baumslag–Solitar groups

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( p,q )) is not finitely generated when p properly divides q .

A formula for the normal subgroup growth of Baumslag–Solitar groups

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.

Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag-Solitar groups

AbstractWe say that a group has property R ∞ if any group automorphism has an infinite number of twisted conjugacy classes. Fel'shtyn and Goncalves proved that the solvable Baumslag-Solitar groups BS(1, m) have property R ∞ . We define a solvable generalization Γ(S) of these groups which is shown to have property R ∞ . It is also shown that property R ∞ is geometric for these groups, that is, any group quasi-isometric to Γ(S) has property R ∞ as well.

Low-pass filters and representations of the Baumslag Solitar group

We analyze representations of the Baumslag Solitar group BS(1, N) = (u,t|utu -1 = t N ) that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.

A context-free and a 1-counter geodesic language for a Baumslag–Solitar group

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.

A Simple Proof of a Theorem of Whyte

Kevin Whyte showed that all Baumslag–Solitar groups BS(p,q) with 1 < p < q are quasi-isometric [Whyte, K., Geom. Funct. Anal. 11 (2001), 1327–1343]. We provide an elementary geometric proof.

The large scale geometry of some metabelian groups

We study the large scale geometry of the upper triangular subgroup of PSL2(Z[ 1 n ]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid with infinite dimensional quasi-isometry group. We generalize our results to a larger class of groups which are metabelian and are higher dimensional analogues of the solvable Baumslag-Solitar groups BS(1, n).