Baumslag Solitar Groups Research Articles

On groups whose cogrowth series is the diagonal of a rational series

We show that if a group contains [Formula: see text] as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalizes a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag–Solitar groups [Formula: see text].

On some planar Baumslag–Solitar actions

Let \mathrm{BS}(1,n)= \langle a,b : a b a^{-1}= b^{n}\rangle be the solvable Baumslag–Solitar group for n \geq 2 . We study representations of \mathrm{BS}(1, n) on the plane by orientation-preserving homeomorphisms, assuming that a acts as a linear map and b as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map h defined by the action of a . In case h is diagonalizable over \mathbb{R} , we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases where h is elliptic or parabolic. Then we give applications to the actions of \mathrm{BS}(1, n) by homeomorphisms of the torus.

On elementary theory of completion of solvable Baumslag–Solitar group

On elementary theory of completion of solvable Baumslag–Solitar group

The isomorphism problem for small-rose generalized Baumslag-Solitar groups

A generalized Baumslag-Solitar group is a finitely generated group that acts on a tree with infinite-cyclic vertex and edge stabilizers. In this paper, we show that the isomorphism problem is solvable for small-rose non-ascending generalized Baumslag-Solitar groups.

Model geometries dominated by locally finite graphs

We study model geometries of finitely generated groups. We show that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We also prove the only such groups of cohomological two are surface groups and generalised Baumslag–Solitar groups.

Restrictions on local embeddability into finite semigroups

We expand the concept of local embeddability into finite structures (LEF) for the class of semigroups with investigations of non-LEF structures, a closely related generalising property of local wrapping of finite structures (LWF) and inverse semigroups. The established results include a description of a family of non-LEF semigroups unifying the bicyclic monoid and Baumslag–Solitar groups and demonstrating that inverse LWF semigroups with finite number of idempotents are LEF.

Geometric realisation of relation modules over aspherical groups

We prove that the direct sums of extensions of scalars of relation modules are geometrically realisable as the second homotopy group of a finite 2-complex. We use this to exhibit a finite 2-complex with fundamental group the (10,15) torus knot group and non-free π2, yielding exotic presentations of a group for which no such examples had previously been known. We conclude by constructing stably free non-free modules over an infinite family of Baumslag-Solitar groups; it remains to determine whether these modules are geometrically realisable by finite 2-complexes.

Incommensurable lattices in Baumslag–Solitar complexes

AbstractThis paper concerns locally finite 2‐complexes that are combinatorial models for the Baumslag–Solitar groups . We show that, in many cases, the locally compact group contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion‐free, and coherent.

Quadratic equations in metabelian Baumslag–Solitar groups

For a finitely generated group G, the Diophantine problem over G is the algorithmic problem of deciding whether a given equation [Formula: see text] (perhaps restricted to a fixed subclass of equations) has a solution in G. In this paper, we investigate the algorithmic complexity of the Diophantine problem for the class [Formula: see text] of quadratic equations over the metabelian Baumslag–Solitar groups [Formula: see text]. We prove that this problem is [Formula: see text]-complete whenever [Formula: see text], and determine the algorithmic complexity for various subclasses (orientable, nonorientable, etc.) of [Formula: see text].

Quasi-isometry invariance of relative filling functions (with an appendix by Ashot Minasyan)

For a finitely generated group G and collection of subgroups \mathcal{P} , we prove that the relative Dehn function of a pair (G,\mathcal{P}) is invariant under quasi-isometry of pairs. Along the way, we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned-off Cayley graphs. We also prove that for a cocompact simply connected combinatorial G - 2 -complex X with finite edge stabilisers, the combinatorial Dehn function is well defined if and only if the 1 -skeleton of X is fine. We also show that if H is a hyperbolically embedded subgroup of a finitely presented group G , then the relative Dehn function of the pair (G, H) is well defined. In the appendix, it is shown that the Baumslag–Solitar group \mathrm{BS}(k,l) has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither k divides l nor l divides k .

Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

AbstractWe look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$ . In the case of $\mathbb {F}_n\times \mathbb {Z}$ , the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

Higher rank confining subsets and hyperbolic actions of solvable groups

Recent papers of the authors have completely described the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace, Cornulier, Monod, and Tessera, which applies, in particular, to solvable groups with virtually cyclic abelianizations. In this paper, we extend this machinery and give a correspondence between the hyperbolic actions of certain solvable groups with higher rank abelianizations and confining subsets of these more general groups. We then apply this extension to give a complete description of the hyperbolic actions of generalized solvable Baumslag-Solitar groups and to reprove a result of Sgobbi-Wong computing their Bieri-Neumann-Strebel invariants.

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.

Baumslag–Solitar groups and residual nilpotence

AbstractLet G be a Baumslag–Solitar group. We calculate the intersection $\gamma_{\omega}(G)$ of all terms of the lower central series of G. Using this, we show that $[\gamma_{\omega}(G),G]=\gamma_{\omega}(G)$, thus answering a question of Bardakov and Neschadim [1]. For any $c \in \mathbb{N}$, with $c \geq 2$, we show, by using Lie algebra methods, that the quotient group $\gamma_{c}(G)/\gamma_{c+1}(G)$ of the lower central series of G is finite.

Knapsack and the power word problem in solvable Baumslag–Solitar groups

We prove that the power word problem for certain metabelian subgroups of [Formula: see text] (including the solvable Baumslag–Solitar groups [Formula: see text]) belongs to the circuit complexity class [Formula: see text]. In the power word problem, the input consists of group elements [Formula: see text] and binary encoded integers [Formula: see text] and it is asked whether [Formula: see text] holds. Moreover, we prove that the knapsack problem for [Formula: see text] is [Formula: see text]-complete. In the knapsack problem, the input consists of group elements [Formula: see text] and it is asked whether the equation [Formula: see text] has a solution in [Formula: see text]. For the more general case of a system of so-called exponent equations, where the exponent variables [Formula: see text] can occur multiple times, we show that solvability is undecidable for [Formula: see text].

Von Staudt constructions for skew-linear and multilinear matroids

This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction, by which we translate our problems to algebra. After giving an exposition of a simple variant of the von Staudt construction we present the following results:Undecidability of several matroid representation problems over division rings. An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic \(0\). An example of a skew-linear matroid that is not multilinear.  Mathematics Subject Classifications: 05B35, 52B40, 14N20, 52C35, 20F10, 03D40Keywords: Matroids, division ring representations, subspace arrangements, \(c\)-arrange\-ments, multilinear matroids, von Staudt constructions, word problem, Weyl algebra, Baumslag-Solitar group

The BNS invariants of the generalized solvable Baumslag-Solitar groups and of their finite index subgroups

We compute the Bieri-Neumann-Strebel invariants for the generalized solvable Baumslag-Solitar groups and their finite index subgroups. Using , we show that certain finite index subgroups of cannot be isomorphic to for any k. In addition, we use the BNS-invariants to give a new proof of property for the groups and their finite index subgroups.

Commensurators of abelian subgroups of biautomatic groups

We show that the commensurator of any finitely generated abelian subgroup H in a biautomatic group centralises a finite-index subgroup of H. We deduce that the CAT(0) groups introduced by Leary and Minasyan (Commensurating HNN-extensions: non-positive curvature and biautomaticity, Geom Topol 25:1819–1860, 2021) are either biautomatic or cannot arise as subgroups of biautomatic groups, answering a question posed in Leary and Minasyan (Commensurating HNN-extensions: non-positive curvature and biautomaticity, Geom Topol 25:1819–1860, 2021) and generalising an analogous result for Baumslag–Solitar groups. These are the first examples of CAT(0) groups that are not subgroups of biautomatic groups.

Embeddings of trees, cantor sets and solvable Baumslag–Solitar groups

We characterise when there exists a quasiisometric embedding between two solvable Baumslag–Solitar groups. This extends the work of Farb and Mosher on quasiisometries between the same groups. More generally, we characterise when there can exist a quasiisometric embedding between two treebolic spaces. This allows us to determine when two treebolic spaces are quasiisometric, confirming a conjecture of Woess. The question of whether there exists a quasiisometric embedding between two treebolic spaces turns out to be equivalent to the question of whether there exists a bilipschitz embedding between two symbolic Cantor sets, which in turn is equivalent to the question of whether there exists a rough isometric embedding between two regular rooted trees. Hence we answer all three of these questions simultaneously. It turns out that the existence of such embeddings is completely determined by the boundedness of an intriguing family of integer sequences.

Regular left-orders on groups

A regular left-order on a finitely generated group G is a total, left-multiplication invariant order on G whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and we give a classification of the groups whose left-orders are all regular left-orders. In addition, we prove that a solvable Baumslag–Solitar group B(1,n) admits a regular left-order if and only if n\geq -1 . Finally, Hermiller and Šunić showed that no free product admits a regular left-order. We show that if A and B are groups with regular left-orders, then (A*B)\times \mathbb{Z} admits a regular left-order.