We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the rooted $d$-regular tree which provide examples of the following interesting phenomena. For $d=2$ and any $m\geq 2$, we get an uncountable family of non quasi-isometric Cayley graphs with the same Laplacian spectrum, absolutely continuous on the union of two intervals, that we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another.
For each $d\geq 3, m\geq 1$, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct moreover a complete system of eigenfunctions which are strongly localized.
We show that if H is a quasiconvex subgroup of a hyperbolic group G then the relative Cayley graph Y (also known as the Schreier coset graph) for G/H is Gromov-hyperbolic. We also observe that in this situation if G is torsion-free and non-elementary and H has infinite index in G then the simple random walk on Y is transient.
We prove that the number of iterations taken by the Weisfeiler-Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs, and a lower bound holds for particular cases, such as for Schreier graphs with $G=\mbox{SL}_{n}({\mathbb F}_{q})$ ($q>2$) acting on $k$-tuples of vectors in ${\mathbb F}_{q}^{n}$; moreover, an exact expression is found in the case of Cayley graphs.
Let the group $G$ act transitively on the finite set $Ω$, and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $Sch(G \circlearrowleft Ω,S)$ is the graph with vertex set $Ω$ and edge set $\{ (ω,ω^s) : ω\in Ω, s \in S \}$. In this paper, we show that random Schreier graphs on $C \log|Ω|$ elements exhibit a (two-sided) spectral gap with high probability, magnifying a well known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of $G$ on $Ω$, we give a lower bound on the number of elements which are necessary to provide a …
We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk's group acting on the boundary of the infinite binary tree. We establish a connection between the action of $G$ on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain s…
We calculate the spectra and spectral measures associated to random walks on restricted wreath products of finite groups with the infinite cyclic 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 Zuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by automata coincides with the Kesten-von Neumann-Serre spectral measure.
We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior.
We give an example of a 4-regular infinite automatic graph of intermediate growth. It is constructed as a Schreier graph of a certain group generated by 3-state automaton. The question was motivated by an open problem on the existence of Cayley automatic groups of intermediate growth.
We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of $f(z)=z^2-1$ on associated coverings of the Riemann sphere. We then derive some algebraic properties of the group, and describe for that specific example the connection between group theory, geometry and dynamics. The most striking is probably that the quotient Cayley graphs of the group (aka ``Schreier graphs'') converge to the Julia set of $f$.
There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-menti…
We prove an inequality on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application we extend Kesten's theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs holds on average. More precisely, we show that if $\mathcal G$ is an infinite deterministic Ramanujan graph, then the time spent in short cycles by a random walk of length $n$ is $o(n)$.
Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are Schreier graphs of the action of the Hanoi Towers group on the levels of the rooted ternary tree. The spectrum of the limiting graph (Schreier graph of the action on the boundary of the tree) is also provided.
The Schreier graphs of Thompson's group F with respect to the stabilizer of 1/2 and generators x_0 and x_1, and of its unitary representation in L_2([0,1]) induced by the standard action on the interval [0,1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x_nv, where n>=0 and v is any word over the alphabet {x_0, x_1}, is constructed. It is proved that the latter graph is non-amenable.
CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)
Institut Curie, CNRS UMR168, Paris, France（居里研究所，法国国家科学研究中心UMR168，法国巴黎）；University of Washington（华盛顿大学）；Three Gorges Mathematical Research Center, China Three Gorges University（三峡数学研究中心，三峡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）；Independent Researcher（独立研究员）；Moscow State University（莫斯科国立大学）；Kazakh-British Technical University（哈萨克-英国技术大学）；Neapolis University Pafos, Cyprus（尼阿波利斯大学帕福斯校区，塞浦路斯）；City St George's, University of London（伦敦大学城市圣乔治学院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；ITMO University（ITMO大学）；Moscow Center for Advanced Studies（莫斯科高等研究中心）；National Research University Higher School of Economics（国立研究大学高等经济学院）；Ariel University（阿里埃勒大学）；Saint-Petersburg Pasteur Institute（圣彼得堡巴斯德研究所）；Criteo AI Lab, Berlin, Germany（Criteo人工智能实验室，德国柏林）；University of Bologna（博洛尼亚大学）；Faculty of Bioengineering and Bioinformatics, Lomonosov Moscow State University（莫斯科国立大学生物工程与生物信息学系）；Researcher at Banner Stat（Banner Stat研究员）；MIREA — Russian Technological University（MIREA — 俄罗斯技术大学）；Research Center of the Artificial Intelligence Institute, Innopolis University（因诺波利斯大学人工智能研究所研究中心）；Inpdependent Researcher（独立研究员）；LIX, Ecole Polytechnique/CNRS, IP Paris（LIX，巴黎综合理工学院/法国国家科学研究中心，巴黎理工学院）
For any order of growth $f(n)=o(\log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function with slower growth. The construction uses permutational wreath products in which the base group is defined via its properly chosen Schreier graph.
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $Γ$, does there exist a generating set $S$ such that the Cayley graph $(Γ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(Γ,S)<p_u(Γ,S)$? We show that this is true if $Γ$ contains an infinite normal subgroup $N$ such that $Γ/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $Γ$ there exists a generating set $S'$ of $G$ such that $p_c(G,S')<p_u(G,S')$. In particular this applies to fr…
An invariant random subgroup of the countable group Γ is a random subgroup of Γ whose distribution is invariant under conjugation by all elements of Γ. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on Γ is strictly less than the spectral radius of the corresponding random walk on Γ/H. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approxim…
Nous montrons la propriété du trou spectral pour la famille des graphes de Cayley obtenus par réduction modulo $q$ d'un sous-groupe de $\mathrm{SL}_d(\mathbb{Z})$ dont l'adhérence de Zariski est un $\mathbb{Q}$-groupe simple. -- We show a spectral gap property for the family of Cayley graphs obtained by reduction modulo $q$ of a subgroup of $\mathrm{SL}_d(\mathbb{Z})$ whose Zariski closure is a simple $\mathbb{Q}$-group.
We provide a self-similar measure for the self-similar group $G$ acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial $z^2+i$. We also provide an $L$-presentation for $G$ and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of $G$ on the orbit of a point on the boundary of the binary rooted tree.
Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.
In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem.
The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a $2n \times 2n$ matrix associated to a $2n$-dimensional permutation representation of $W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous' spectral gap conjecture', proven in 2010 by Caputo, Liggett and Richthammer.
Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$ are adjacent if and only if $ab^{-1}\in S$. The adjacency spectrum $Spec(Γ)$ of a graph $Γ$ is the multiset of eigenvalues of its adjacency matrix. A graph $Γ$ is called "determined by its spectrum" (or for short DS) whenever if a graph $Γ'$ has the same spectrum as $Γ$, then $Γ\cong Γ'$. We say that the group $G$ is DS (Cay-DS, respectively) whenever if $Γ$…
Let $S_n$ denote the symmetric group on $n$ elements, and $Σ\subseteq S_{n}$ a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $Σ$ is a set of transpositions, then the second eigenvalue of the Cayley graph $\mathrm{Cay}\left(S_{n},Σ\right)$ is identical to the second eigenvalue of the Schreier graph on $n$ vertices depicting the action of $S_{n}$ on $\left\{ 1,\ldots,n\right\}$. Inspired by this seminal result, we study similar questions for other types of sets in $S_{n}$. Specifically, we consider n…
For each $p\geq 1$, the star automaton group $\mathcal{G}_{S_p}$ is an automaton group which can be defined starting from a star graph on $p+1$ vertices. We study Schreier graphs associated with the action of the group $\mathcal{G}_{S_p}$ on the regular rooted tree $T_{p+1}$ of degree $p+1$ and on its boundary $\partial T_{p+1}$. With the transitive action on the $n$-th level of $T_{p+1}$ is associated a finite Schreier graph $Γ^p_n$, whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as limits of the sequence $…
We describe the infinite family of spider-web graphs $S_{k,M,N }$, $k \geq 2$, $M \geq 1$ and $N \geq 0$, studied in physical literature as tensor products of well-known de Brujin graphs $B_{k,N}$ and cyclic graphs $C_M$ and show that these graphs are Schreier graphs of the lamplighter groups $L_k = Z/kZ \wr Z$. This allows us to compute their spectra and to identify the infinite limit of $S_{k,M,N}$, as $N, M \to\infty$, with the Cayley graph of the lamplighter group $L_k$. This is the final version of the article, taking in account comments from the referees and with an extended introduction…
Let $G$ be a group and $S\subseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}=\{s^{-1}\mid s\in S\}$. Then {\it the Cayley graph ${\rm Cay}(G,S)$} is an undirected graph $Γ$ with the vertex set $V(Γ)=G$ and the edge set $E(Γ)=\{(g,gs)\mid g\in G, s\in S\}$. A graph $Γ$ is said to be {\it integral} if every eigenvalue of the adjacency matrix of $Γ$ is integer. In the paper, we prove the following theorem: {\it if a subset $S=S^{-1}$ of $G$ is normal and $s\in S\Rightarrow s^k\in S$ for every $k\in \mathbb{Z}$ such that $(k,|s|)=1$, then ${\rm Cay}(G,S)$ is integral.} In particular, {\it …
We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also fo…
The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this paper is to begin to develop such a framework for $k$-regular simplicial complexes of general dimension $d$. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of $k$-regular simplicial complexes as quotients of one universal object: the $k$-regular $d$-dimensional arboreal complex, which is its…
Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.
A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schrödinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayle…
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 …
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1…
This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of ``prime cycles'' at a vertex. The bounds are obtained by constructing circuits in the graph that resemble ``cactus trees'', and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence and underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons…
