Lamplighter Group Research Articles

On invariant generating sets for the cycle space

Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop O ( 1 ) O(1) model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop O ( 1 ) O(1) model.

On reversible automata generating lamplighter groups

For any nontrivial abelian group X we construct a reversible automaton such that its set of states and alphabet are identified with X, transition and output functions are defined via the left regular action and its group splits into the restricted wreath product X≀Z, i.e. is a lamplighter group.

The degree of commutativity of wreath products with infinite cyclic top group

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups G: the degree of commutativity dcS(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G)$$\\end{document}, with respect to a given finite generating set S, results from considering the fractions of commuting pairs of elements in increasing balls around 1G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1_G$$\\end{document} in the Cayley graph . We focus on restricted wreath products of the form G=H≀⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G = H \\hspace{1.111pt}{\\wr }\\hspace{1.111pt}\\langle \\hspace{1.111pt}t \\rangle $$\\end{document}, where H≠1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \ e 1$$\\end{document} is finitely generated and the top group ⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\hspace{1.111pt}t \\rangle $$\\end{document} is infinite cyclic. In accordance with a more general conjecture, we show that dcS(G)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G) = 0$$\\end{document} for such groups G, regardless of the choice of S. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox’s main auxiliary result: in ‘reasonably large’ homomorphic images of wreath products G as above, the image of the base group has density zero, with respect to certain types of generating sets.

Spectral dynamics for the infinite dihedral group and the lamplighter group

Spectral dynamics for the infinite dihedral group and the lamplighter group

On Bohr compactifications and profinite completions of group extensions

AbstractLet $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.

Dead ends on wreath products and lamplighter groups

For any finite group A and any finitely generated group B, we prove that the corresponding lamplighter group [Formula: see text] admits a standard generating set with unbounded depth, and that if B is abelian then the above is true for every standard generating set. This generalizes the case where [Formula: see text] together with its cyclic generator [S. Cleary and J. Taback, Dead end words in lamplighter groups and other wreath products, Q. J. Math. 56(2) (2005) 165–178]. When [Formula: see text] is the free product of two finite groups H and K, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

Self-Similar Groups and Holomorphic Dynamics: Renormalization, Integrability, and Spectrum

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

Quantifying conjugacy separability in wreath products of groups

Abstract We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of A and B, the growth of $\mathcal{C}$-cyclic subgroup separability of B and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterization of when $A \wr B$ is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.

Simulations and the lamplighter group

We introduce a notion of “simulation” for labelled graphs, in which edges of the simulated graph are realized by regular expressions in the simulating graph, and we prove that the tiling problem (a.k.a. the “domino problem”) for the simulating graph is at least as difficult as that for the simulated graph. We apply this to the Cayley graph of the “lamplighter group” L=\mathbb Z/2\wr\mathbb Z , and more generally to “Diestel–Leader graphs”. We prove that these graphs simulate the plane, and thus deduce that the seeded tiling problem is unsolvable on the group L . We note that L does not contain any plane in its Cayley graph, so our undecidability criterion by simulation covers cases not addressed by Jeandel’s criterion based on translation-like action of a product of finitely generated infinite groups. Our approach to tiling problems is strongly based on categorical constructions in graph theory.

Lamplighter groups and automata

Lamplighter groups and automata

Approximating the group algebra of the lamplighter by infinite matrix products

AbstractIn this paper, we introduce a new technique in the study of the*{*}-regular closure of some specific group algebrasKGinside𝒰⁢(G){{\mathcal{U}}(G)}, the*{*}-algebra of unbounded operators affiliated to the group von Neumann algebra𝒩⁢(G){{\mathcal{N}}(G)}. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the formCK⁢(X)⋊Tℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}, whereXis a totally disconnected compact metrizable space,Tis a homeomorphism ofX, andCK⁢(X){C_{K}(X)}stands for the algebra of locally constant functions onXwith values on an arbitrary fieldK. The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family ofℓ2{\ell^{2}}-Betti numbers arising from the lamplighter group, most of them transcendental.

Lamplighters and the bounded cohomology of Thompson’s group

We prove the vanishing of the bounded cohomology of lamplighter groups for a wide range of coefficients. This implies the same vanishing for a number of groups with self-similarity properties, such as Thompson’s group F. In particular, these groups are boundedly acyclic. Our method is ergodic and applies to “large” transformation groups where the Mather–Matsumoto–Morita method sometimes fails because not all are acyclic in the usual sense.

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.

Stochastic approximation of lamplighter metrics

We observe that embeddings into random metrics can be fruitfully used to study the L 1 $L_1$ -embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L 1 $L_1$ -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n $n$ -point metric space embeds bi-Lipschitzly into L 1 $L_1$ with distortion O ( log n ) $O(\log n)$ . In particular, for every finite group G $G$ the lamplighter group H = Z 2 ≀ G $H=\mathbb {Z}_2\wr G$ bi-Lipschitzly embeds into L 1 $L_1$ with distortion O ( log log | H | ) $O(\log \log |H|)$ . In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L 1 $L_1$ of the lamplighter group over the d $d$ -dimensional infinite lattice Z d $\mathbb {Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d $d$ -dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

SOLENOIDAL MAPS, AUTOMATIC SEQUENCES, VAN DER PUT SERIES, AND MEALY AUTOMATA

AbstractThe ring$\mathbb Z_{d}$ofd-adic integers has a natural interpretation as the boundary of a rootedd-ary tree$T_{d}$. Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from$\mathbb Z_{d}$to itself. In the case when$d=p$is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl.4(2) (2012), 151–160] showed that$f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute ap-automatic sequence over a finite subset of$\mathbb Z_{p}\cap \mathbb Q$. We generalize this result to arbitrary integers$d\geq 2$and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other andvice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.

The Homology of the Lamplighter Lie Algebra

We show that the associated Lie algebra of the Malcev $\mathbb{Q}$-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. We also prove that the homology of this completed Lie algebra is of uncountable dimension on each degree.

The algebraic solvability of the novel lamplighter puzzle

In this study, the group of finite cyclic lamplighter states is reinterpreted as the novel lamplighter puzzle. The rules of the puzzle are outlined and related back to properties of the lamplighter group with specific interest placed upon the discussion of which puzzle instances are solvable. The paper shows that, through the use of algebra, many puzzle instances can be identified as solvable without the use of an exhaustive search algorithm. Solvability depends upon the creation of irregular generating sets for subgroups of the finite cyclic lamplighter group and the cosets formed by these subgroups. Further possible generalizations of the lamplighter puzzle are also discussed in closing.

Law of large numbers for the drift of the two-dimensional wreath product

We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite (2+epsilon )-moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is [0,infty ). We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

Random walks on the discrete affine group

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.