Diestel-Leader Graphs Research Articles

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.

On the asymmetry of stars at infinity

Given a bordified space, Karlsson defines an incidence geometry of stars at infinity. These stars and their incidence are closely related to well-understood objects when the space is hyperbolic, CAT(0), or a bounded convex domain with the Hilbert metric. A question stemming from Karlsson's original paper was whether or not the relation of one boundary point being included in a star of another boundary point is symmetric. This paper provides an example demonstrating that this relation in the star boundary of the three-tree Diestel-Leader graph DL3(q) is not symmetric. In doing so, some interesting bounds on distance in Diestel-Leader graphs are utilized.

Shannon’s theorem for locally compact groups

We establish the ray and strip approximation criteria for the identification of the Poisson boundary of random walks on locally compact groups. This settles a conjecture from the 1990s by Kaimanovich, who formulated and proved the criterion for discrete groups. The key result is the proof of a version of the Shannon–McMillan–Breiman theorem for locally compact groups. We provide several applications to locally compact groups of isometries of nonpositively curved spaces, as well as Diestel–Leader graphs and horocylic products.

Lamplighter groups, de Brujin graphs, spider-web graphs and their spectra

We study the infinite family of spider-web graphs , , and , initiated in the 50s in the context of network theory. It was later shown in physical literature that these graphs have remarkable percolation and spectral properties. We provide a mathematical explanation of these properties by putting the spider-web graphs in the context of group theory and algebraic graph theory. Namely, we realize them as tensor products of the well-known de Bruijn graphs with cyclic graphs and show that these graphs are described by the action of the lamplighter group on the infinite binary tree. Our main result is the identification of the infinite limit of , as , with the Cayley graph of the lamplighter group which, in turn, is one of the famous Diestel–Leader graphs . As an application we compute the spectra of all spider-web graphs and show their convergence to the discrete spectral distribution associated with the Laplacian on the lamplighter group.

Automorphisms of higher rank lamplighter groups

Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].

Metric properties of Diestel-Leader groups

In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$, as well as infinitely many cone types and hence no regular language of geodesics. These results are proven using a combinatorial formula to compute the word length of group elements with respect to $S_{d,q}$ which is also proven in the paper and relies on the geometry of the Diestel-Leader graphs.

Statistical hyperbolicity in groups

In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (i.e., without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products, for Diestel-Leader graphs and lamplighter groups. In free abelian groups, the word metrics asymptotically approach norms induced by convex polytopes, causing the study of sprawl to reduce to a problem in convex geometry. We present an algorithm that computes sprawl exactly for any generating set, thus quantifying the failure of various presentations of Z^d to be hyperbolic. This leads to a conjecture about the extreme values, with a connection to the classic Mahler conjecture.

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.

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.

Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel–Leader graph DL ( q , r ) , where q , r ⩾ 2 . The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1 . When q = r , it is the Cayley graph of the wreath product (lamplighter group) Z q ≀ Z with respect to a natural set of generators. We describe the full Martin compactification of these random walks on DL -graphs and, in particular, lamplighter groups. This completes previous results of Woess, who has determined all minimal positive harmonic functions.

Lamplighters, Diestel–Leader Graphs, Random Walks, and Harmonic Functions

The lamplighter group over -graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive harmonic functions.

Spectral Computations on Lamplighter Groups and Diestel-Leader Graphs

The Diestel-Leader graph DL(q,r) is the horocyclic product of the homogeneous trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of the lamplighter group (wreath product of the cyclic group of order q with the infinite cyclic group) with respect to a natural generating set. For the "Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and Dicks & Schick have determined the spectrum and the (on-diagonal) spectral measure (Plancherel measure). Here, we show that thanks to the geometric realization, these results can be obtained for all DL-graphs by directly computing an l^2-complete orthonormal system of finitely supported eigenfunctions of the SRW. This allows computation of all matrix elements of the spectral resolution, including the Plancherel measure. As one application, we determine the sharp asymptotic behaviour of the N-step return probabilities of SRW. The spectral computations involve a natural approximating sequence of finite subgraphs, and we study the question whether the cumulative spectral distributions of the latter converge weakly to the Plancherel measure. To this end, we provide a general result regarding Foelner approximations; in the specific case of DL(q,r), the answer is positive only when r=q.

Boundary and Entropy of Space Homogeneous Markov Chains

We study the Poisson boundary ($\equiv$ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations. This automorphism group is locally compact, but not necessarily discrete or unimodular. The main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups, while, however, the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other. Various new examples are given as applications, including a description of the Poisson boundary for random walks on vertex-transitive graphs with infinitely many ends and on the Diestel-Leader graphs.