Baumslag Solitar Groups Research Articles

Vertex separators, chordality and virtually free groups

In this paper, we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric group theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumslag–Solitar group [Formula: see text] is [Formula: see text]-chordal for some [Formula: see text] if and only if [Formula: see text] and we give an application of generalized chordality to the study of the word problem.

Train track maps on graphs of groups

In this paper we develop the theory of train track maps on graphs of groups. Expanding a definition of Bass, we define a notion of a map of a graph of groups, and of a homotopy equivalence. We prove that under one of two technical hypotheses, any homotopy equivalence of a graph of groups may be represented by a relative train track map. The first applies in particular to graphs of groups with finite edge groups, while the second applies in particular to certain generalized Baumslag–Solitar groups.

Poincaré profiles of Lie groups and a coarse geometric dichotomy

Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form Ntimes S, where N is a connected nilpotent Lie group, and S is a rank one simple Lie group, both the growth exponent of N, and the conformal dimension of S are non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder.

Group and Algorithmic Properties of Generalized Baumslag–Solitar Groups

Group and Algorithmic Properties of Generalized Baumslag–Solitar Groups

Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi- $$\kappa $$ -to-one in a natural sense for some $$\kappa >0$$ . For non-amenable graphs, all quasi-isometries are quasi- $$\kappa $$ -to-one for any $$\kappa >0$$ , while for amenable ones there exists at most one possible such $$\kappa $$ . For an amenable graph X, we show that the set of possible $$\kappa $$ forms a subgroup of $$\mathbb {R}_{>0}$$ that we call the (measure-)scaling group of X. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a given locally compact group have same scaling groups. We compute the scaling group in a number of cases. For instance it is all of $$\mathbb {R}_{>0}$$ for lattices in Carnot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup $$\mathbb {Q}_{>0}$$ for lamplighter groups over finitely presented amenable groups.

Aperiodic SFTs on Baumslag-Solitar groups

We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups. We show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike Z2, but like Z3, strong and weak aperiodic SFTs are different classes of SFTs in residually finite BS groups. More precisely, we prove that a weakly aperiodic SFT on BS(m,n) due to Aubrun and Kari is, in fact, strongly aperiodic on BS(1,n); and weakly but not strongly aperiodic on any other BS(m,n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS(1,n); and we show that there exists a strongly aperiodic SFT on BS(n,n).

A new proof of the growth rate of the solvable Baumslag–Solitar groups

We exhibit a regular language of geodesics for a large set of elements of BS(1, n) and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of BS(1, n), which was initially computed by Collins et al. (AM (Basel) 62:1-11, 1994). Our methods are based on those we develop in Taback and Walker (JTA, to appear) to show that BS(1, n) has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan et al. (JTA, 2020, https://doi.org/10.1142/S1793525321500096 ).

Negative immersions for one-relator groups

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be a free group, and let w∈F be a nonprimitive element. The primitivity rank of w, π(w), is the smallest rank of a subgroup of F containing w as an imprimitive element. Then any subgroup of the one-relator group G=F∕⟨⟨w⟩⟩ generated by fewer than π(w) elements is free. In particular, if π(w)>2, then G does not contain any Baumslag–Solitar groups. The hypothesis that π(w)>2 implies that the presentation complex X of the one-relator group G has negative immersions: if a compact, connected complex Y immerses into X and χ(Y)≥0, then Y Nielsen reduces to a graph. The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus’ Freiheitssatz and theorems of Lyndon, Baumslag, Stallings, and Duncan–Howie. The dependence theorem strengthens Wise’s w-cycles conjecture, proved independently by the authors and Helfer–Wise, which implies that the one-relator complex X has nonpositive immersions when π(w)>1.

The Ribes-Zalesskii property of some one relator groups

The profinite topology on any abstract group $G$, is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group $G$ has the Ribes-Zalesskii property of rank $k$, or is RZ$_{k}$ with $k$ a natural number, if any product $H_{1} H_{2} \cdots H_{k}$ of finitely generated subgroups $H_{1}, H_{2}, \cdots , H_{k}$ is closed in the profinite topology on $G$. And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ$_{k}$ for any natural number $k$. In this paper we characterize groups which are RZ$_{2}$. Consequently, we obtain condition under which a free product with amalgamation of two RZ$_{2}$ groups is RZ$_{2}$. After observing that the Baumslag-Solitar groups $BS (m, n)$ are RZ$_{2}$ and clearly RZ if $m= n$, we establish some suitable properties on the RZ$_{2}$ property for the case when $m= -n$. Finally, since any group $BS (m, n)$ can be viewed as a HNN-extension, then we point out the Ribes-Zalesskii property of rank two on some HNN-extensions.

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.

Isomorphism classification of Leary–Minasyan groups

Abstract Recently, I. J. Leary and A. Minasyan [Commensurating HNN extensions: Nonpositive curvature and biautomaticity, Geom. Topol. 25 (2021), 4, 1819–1860] studied the class of groups G ⁢ ( A , L ) G(A,L) defined as commensurating HNN-extensions of Z n \mathbb{Z}^{n} . This class, containing the class of Baumslag–Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups G ⁢ ( A , L ) G(A,L) up to isomorphism.

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

AbstractFor an ascending HNN-extension$G*_{\psi }$of a finitely generated abelian groupG, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in$\mathcal {A}^{G*_{\psi }}$forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups$\mathrm {BS}(1,N)$,$N\ge 2$, for which our results imply that a$\mathrm {BS}(1,N)$-subshift of finite type which contains a configuration with period$a^{N^\ell }\!, \ell \ge 0$, must contain a strongly periodic configuration with monochromatic$\mathbb {Z}$-sections. Then we study propern-colorings,$n\ge 3$, of the (right) Cayley graph of$\mathrm {BS}(1,N)$, estimating the entropy of the associated subshift together with its mixing properties. We prove that$\mathrm {BS}(1,N)$admits a frozenn-coloring if and only if$n=3$. We finally suggest generalizations of the latter results ton-colorings of ascending HNN-extensions of finitely generated abelian groups.

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).

Characterisation of homotopy ribbon discs

Let Γ be either the infinite cyclic group Z or the Baumslag-Solitar group Z⋉Z[12]. Let K be a slice knot admitting a slice disc D in the 4-ball whose exterior has fundamental group Γ. We classify the Γ-homotopy ribbon slice discs for K up to topological ambient isotopy rel. boundary. In the infinite cyclic case, there is a unique equivalence class of such slice discs. When Γ is the Baumslag-Solitar group, there are at most two equivalence classes of Γ-homotopy ribbon discs, and at most one such slice disc for each lagrangian of the Blanchfield pairing of K.

Uncountable groups and the geometry of inverse limits of coverings

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor lim←1, which is, however, notoriously difficult to compute when the fundamental group, π1(X), is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit X˜ of a sequence of covering spaces in terms of the functors lim← and lim←1 applied to sequences of countable groups arising from polyhedral approximations of X.A consequence of our computation is that path-connectedness of a lifting space, X˜, implies that π1(X˜) supplements π1(X) in πˇ1(X) where πˇ1(X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that G⋅KerZ(Fˆ)=Fˆ≠G⋅KerB(1,n)(Fˆ), where Fˆ is the canonical inverse limit of finite rank free groups, G is the fundamental group of the Hawaiian Earring, B(1,n) is the Baumslag-Solitar group, and KerA(Fˆ) is the intersection of kernels of homomorphisms from Fˆ to A.

Certain residual properties of generalized Baumslag–Solitar groups

Let G be a generalized Baumslag–Solitar group, and let C be a class of groups containing at least one non-trivial group and closed under taking subgroups, extensions, and Cartesian products of the form ∏y∈YXy, where X, Y∈C and Xy is an isomorphic copy of X for every y∈Y. We give a criterion for G to be residually a C-group provided C consists only of periodic groups. We also prove that G is residually a torsion-free C-group if C contains at least one non-periodic group and is closed under taking homomorphic images. These facts generalize and strengthen some known results. We also provide criteria for a GBS-group to be a) residually nilpotent; b) residually torsion-free nilpotent; c) residually free.

A closer look at the non-Hopfianness of $BS(2,3)$

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, that is not injective. In particular this is equivalent to saying that $BS(2,3)$ has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of $BS(2,3)$ has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism $\phi$. We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how $\phi$ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt.

Finite index subgroups in non-large generalized Baumslag–Solitar groups

A generalized Baumslag–Solitar group (GBS-group) is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. A group is large if some finite index subgroup maps onto the free group F 2. An explicit description is given for all index n subgroups in non-large GBS groups. Moreover, we give exact formulas for the numbers of subgroups, normal subgroups, and conjugacy classes of subgroups of each finite index in non-large GBS groups.

Generating the inverse limit of free groups

We study the relation between two uncountable groups with remarkable properties (cf. [15]): the topological free product of infinite cyclic groups G (the fundamental group of the Hawaiian Earring), and the inverse limit of finitely generated free groups Fˆ. The former has a canonical embedding as a proper subgroup of the latter and we examine when G, together with certain naturally defined normal subgroups of Fˆ generate the entire group Fˆ. We are interested in particular in normal subgroupsKerT(Fˆ)=⋂{Kerφ|φ∈hom⁡(Fˆ,T)}, where T is some finitely-presented n-slender group.Our main results state that if T is the infinite cyclic group or the free nilpotent class 2 group on 2 generators, then G and KerT(Fˆ) generate Fˆ. On the other hand, if T is the free nilpotent class 3 group or a Baumslag-Solitar group, then the product of subgroups G⋅KerTFˆ is a proper subgroup of Fˆ.In the last section, we provide an interesting geometric interpretation of the above results in terms of path-connectedness of certain fibrations arising as inverse limits of covering spaces over the Hawaiian earring space.

A Cuntz-Pimsner model for the C⁎-algebra of a graph of groups

We provide a Cuntz-Pimsner model for graph of groups C⁎-algebras. This allows us to compute the K-theory of a range of examples and show that graph of groups C⁎-algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincaré duality in KK-theory. By constructing a K-theory duality class we compute the K-homology of these crossed products.