Lamplighter Group Research Articles

METABELIAN GROUPS WITH QUADRATIC DEHN FUNCTION AND BAUMSLAG–SOLITAR GROUPS

We prove that groups in a certain class of metabelian locally compact groups, have quadratic Dehn function. As an application, we embed the solvable Baumslag-Solitar groups into finitely presented metabelian groups with quadratic Dehn function. Also, we prove that Baumslag's finitely presented metabelian groups, in which the lamplighter groups embed, have quadratic Dehn function.

Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups

We show that certain lamplighter groups that are quasi-isometric to each other are not bilipschitz equivalent. This gives a positive answer to a question in Topics in Geometric Group Theory by Pierre de la Harpe (page 107).

Tilings and Submonoids of Metabelian Groups

In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product ℤ≀(ℤ×ℤ). We also show that subsemimodule membership is undecidable for finite rank free (ℤ×ℤ)-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.

Groups, graphs and trees: an introduction to the geometry of infinite groups

Preface 1. Cayley's theorems 2. Groups generated by reflections 3. Groups acting on trees 4. Baumslag-Solitar groups 5. Words and Dehn's word problem 6. A finitely-generated, infinite, Torsion group 7. Regular languages and normal forms 8. The Lamplighter group 9. The geometry of infinite groups 10. Thompson's group 11. The large-scale geometry of groups Bibliography Index.

The Euclidean Distortion of the Lamplighter Group

We show that the cyclic lamplighter group C 2 ≀ C n embeds into Hilbert space with distortion \(\mathrm{O}(\sqrt{\log n})\) . This matches the lower bound proved by Lee et al. (Geom. Funct. Anal., 2009), answering a question posed in that paper. Thus, the Euclidean distortion of C 2 ≀ C n is \(\varTheta(\sqrt{\log n})\) . Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni et al. (Isr. J. Math. 52(3):251–265, 1985) and by Gromov (see de Cornulier et. al. in Geom. Funct. Anal., 2009), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.

A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $\Gamma$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich. A geometrical method for constructing the strip as a subset of the lamplighter group starting with a smaller strip in the base group $\Gamma$ is developed. Then, this method is applied to several classes of base groups $\Gamma$: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.

Limits of Baumslag–Solitar groups

We give a parametrization by m-adic integers of the limits of Baumslag–Solitar groups (marked with a canonical set of generators). It is shown to be continuous and injective on the invertible m-adic integers. We show that all such limits are extensions of a free group by a lamplighter group and all but possibly one are not finitely presented. Finally, we give presentations related to natural actions on trees.

On the spectrum of lamplighter groups and percolation clusters

Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group of "lamps" $H$. We show that the spectral measure (Plancherel measure) of any symmetric "switch--walk--switch" random walk on $H \wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilites on the percolation cluster. In particular, if the clusters of percolation with parameter $p$ are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and Zuk, resp. Dicks and Schick regarding the case when $G$ is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter $p$ is always related with the Plancherel measure of a convolution operator by a signed measure on $H \wr G$, where $H = Z$ or another suitable group.

Asymptotic dimension and uniform embeddings

We study uniform embeddings of metric spaces, which satisfy some asymptotic tameness conditions such as finite asymptotic dimension, finite Assouad–Nagata dimension, polynomial dimension growth or polynomial growth, into function spaces. We show how the type function of a space with finite asymptotic dimension estimates its Hilbert (or any ℓ^p -) compression. In particular, we show that the spaces of finite asymptotic dimension with linear type (spaces with finite Assouad–Nagata dimension) have compression rate equal to one. We show, without the extra assumption that the space has the doubling property (finite Assouad dimension), that a space with polynomial growth has polynomial dimension growth and compression rate equal to one. The method used allows us to obtain a lower bound on the compression of the lamplighter group ℤ ≀ ℤ , which has infinite asymptotic dimension.

Proper actions of lamplighter groups associated with free groups

Given a finite group H and a free group F n , we prove that the wreath product H ≀ F n admits a metrically proper, isometric action on a Hilbert space. To cite this article: Y. de Cornulier et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Embeddings of Discrete Groups and the Speed of Random Walks

Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric d G . For a Banach space X, let α * X (G) (respectively, α / # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively, an equivariant mapping) f:G → X and c > 0 such that for all x, y ∈ G we have ‖ f(x) − f(y)‖ ≥ c · d G (x, y) α . In particular, the Hilbert compression exponent (respectively, the equivariant Hilbert compression exponent) of G is (respectively, ). We show that if X has modulus of smoothness of power type p, then . Here β * (G) is the largest β ≥ 0 for which there exists a set of generators S of G and c > 0, such that for all we have , where { W t } ∞ t=0 is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X = L p , generalizes a theorem of Guentner and Kaminker (20), and answers a question posed by Tessera (37). We also show that, if then . This improves the previous bound due to Stalder and Valette (36). We deduce that if we write and then and use this result to answer a question posed by Tessera in (37) on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C 2 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor, and Peres in (26). Finally, we use these results to show that edge Markov type need not imply Enflo type.

On quasi-isometric embeddings of Lamplighter groups

We denote by Γ G \Gamma _G the Lamplighter group of a finite group G G . In this article, we show that if G G and H H are two finite groups with at least two elements, then there exists a quasi-isometric embedding from Γ G \Gamma _G to Γ H \Gamma _H . We also prove that the quasi-isometry group Q I ( Γ G ) {\mathcal Q}I(\Gamma _G) of Γ G \Gamma _G contains all finite groups. We then show that the group of automorphisms of Γ Z n \Gamma _{{\mathbb Z}_n} has infinite index in Q I ( Γ Z n ) {\mathcal Q}I(\Gamma _{{\mathbb Z}_n}) .

Some limits related to random iterations of a lamplighter group

We obtain limits for random iterations of the lamplighter group Z 2 ≀ Z , which refer to the state of a lamp (light “on” or “off”), and the number of changes in a set of lamps.

A finitely-presented solvable group with a small quasi-isometry group

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large-scale geometry of lamplighter groups.

Quasi-isometries and Rigidity of Solvable Groups

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as coarse differentiation.

The Poisson boundary of lamplighter random walks on trees

Let $${\mathbb{T}_q}$$ be the homogeneous tree with degree q + 1 ≥ 3 and $${\mathcal{F}}$$ a finitely generated group whose Cayley graph is $${\mathbb{T}_q}$$ . The associated lamplighter group is the wreath product $${\mathcal{L} \wr \mathcal{F}}$$ , where $${{\mathcal{L}}}$$ is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter “operates at bounded range”.

Cutsets in Infinite Graphs

We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1-ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n$-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $\mathbb{Z}$.

The spectra of lamplighter groups and Cayley machines

We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr \(\mathbb{Z}\), with G a finite group, by calculating the Kesten—von Neumann—Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Żuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by an automaton coincides with the Kesten—von Neumann—Serre spectral measure.

Positive Harmonic Functions for Semi-Isotropic Random Walks on Trees, Lamplighter Groups, and DL-Graphs

We determine all positive harmonic functions for a large class of “semi-isotropic” random walks on the lamplighter group, i.e., the wreath product ℤq≀ℤ, where q≥2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel–Leader graph \(\mathsf{DL}(q,q)\) . More generally, \(\mathsf{DL}(q,r)\) (q,r≥2) is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1, and our result applies to all \(\mathsf {DL}\) -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees.

Horocyclic products of trees

Let T_1,\dots, T_d be homogeneous trees with degrees q_1+1, \dots, q_d+1 \ge 3, respectively. For each tree, let \mathfrak h:T_j \to \mathbb Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T_1,\dots, T_d is the graph \mathsf{DL}(q_1,\dots,q_d) consisting of all d -tuples x_1 \cdots x_d \in T_1 \times \dots \times T_d with \mathfrak h(x_1)+\dots+\mathfrak h(x_d)=0 , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d=2 and q_1=q_2=q then we obtain a Cayley graph of the lamplighter group (wreath product) \mathfrak Z_q \wr \mathbb Z . If d = 3 and q_1 = q_2 = q_3 = q then \mathsf{DL} is the Cayley graph of a finitely presented group into which the lamplighter group embeds naturally. Also when d\ge 4 and q_1 = \dots = q_d = q is such that each prime power in the decomposition of q is larger than d-1 , we show that \mathsf{DL} is a Cayley graph of a finitely presented group. This group is of type F_{d-1} , but not F_d . It is not automatic, but it is an automata group in most cases. On the other hand, when the q_j do not all coincide, \mathsf{DL}(q_1,\dots,q_d) is a vertex-transitive graph, but is not the Cayley graph of a finitely generated group. Indeed, it does not even admit a group action with finitely many orbits and finite point stabilizers. The \ell^2 -spectrum of the “simple random walk” operator on \mathsf{DL} is always pure point. When d=2 , it is known explicitly from previous work, while for d=3 we compute it explicitly. Finally, we determine the Poisson boundary of a large class of group-invariant random walks on \mathsf{DL} . It coincides with a part of the geometric boundary of \mathsf{DL} .