Self-similar Groups Research Articles

COSET ENUMERATION FOR CERTAIN INFINITELY PRESENTED GROUPS

We describe an algorithm that computes the index of a finitely generated subgroup in a finitely L-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely L-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group, and the Hanoi 3-group.

MODIFIED HANOI TOWERS GROUPS AND LIMIT SPACES

We introduce the k-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and give conditions for them to be contracting. We analyze the limit spaces of a particular family of contracting Hanoi groups, [Formula: see text], and show that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, we provide partial results on the contraction of Hanoi groups with weaker symmetry.

Some topics in the dynamics of group actions on rooted trees

This article combines the features of a survey and a research paper. It presents a review of some results obtained during the last decade in problems related to the dynamics of branch and self-similar groups on the boundary of a spherically homogeneous rooted tree and to the combinatorics and asymptotic properties of Schreier graphs associated with a group or with its action. Special emphasis is placed on the study of essentially free actions of selfsimilar groups, which are antipodes to branch actions. At the same time, the theme “free versus nonfree” runs through the paper. Sufficient conditions are obtained for the essential freeness of an action of a self-similar group on the boundary of a tree. Specific examples of such actions are given. Constructions of the associated dynamical system and the Schreier dynamical system generated by a Schreier graph are presented. For groups acting on trees, a trace on the associated C*-algebra generated by a Koopman representation is introduced, and its role in the study of von Neumann factors, the spectral properties of groups, Schreier graphs, and elements of the associated C*-algebra is demonstrated. The concepts of asymptotic expander and asymptotic Ramanujan graph are introduced, and examples of such graphs are given. Questions related to the notion of the cost of action and the notion of rank gradient are discussed.

On the closure of the first Grigorchuk group

We consider the closure (in the natural topology) of the well-known periodic self-similar group. A finitely generated group that is not isomorphic to the Grigorchuk group and has the same closure is constructed.

Scale-invariant groups

Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G_n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups \mathrm{F}\wr \mathbb{Z} , where \mathrm{F} is any finite Abelian group; the solvable Baumslag–Solitar groups \mathrm{BS}(1,m) ; the affine groups A ⋉ \mathbb{Z}^d , for any A\leq \mathrm{GL}(\mathbb{Z},d) . However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Let G be a group and ϕ : H → G be a contracting homomorphism from a subgroup H < G of finite index. V. Nekrashevych (2005) [25] associated with the pair ( G , ϕ ) the limit dynamical system ( J G , s ) and the limit G -space X G together with the covering ⋃ g ∈ G T ⋅ g by the tile T . We develop the theory of self-similar measures m on these limit spaces. It is shown that ( J G , s , m ) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles T ∩ ( T ⋅ g ) for g ∈ G . We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.

A hybrid FPAB/BBO Algorithm for Satellite Image Classification

In the past years, remote sensing has been used for the classification of satellite image on a very large scale. This paper deals with image classification by using swarm computing technique. In this work, we use a new swarm data clustering method based upon flower pollination by artificial bees to cluster the satellite image pixels. The aim of clustering is to separate a set of data points into self-similar groups. Those clusters will be further classified using Biogeography Based Optimization. The results indicate that highly accurate classification of the satellite image is obtained by using the proposed algorithm.

Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients

AbstractWe describe an algorithm for computing successive quotients of the Schur multiplierM(G) of a groupGgiven by an invariant finiteL-presentation. As applications, we investigate the Schur multipliers of various self-similar groups, including the Grigorchuk super-group, the generalized Fabrykowski–Gupta groups, the Basilica group and the Brunner–Sidki–Vieira group.

Relational Generalizations of Cluster Validity Indices

Numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another. One such scheme, clustering, looks for self-similar groups of objects. To use clustering algorithms, an investigator must often have a priori knowledge of the number of clusters, i.e., c, to search for in the data. Moreover, it is often convenient to have ways to rank the returned results, either for a single value of c, a range of c's different clustering methods, or any combination thereof. However, the task of assessing the quality of the results, so that c may be determined objectively, is currently ill-defined for object-object relationships. To bridge this gap, we generalize three well-known validity indices: the modified Hubert's Gamma, Xie-Beni, and the generalized Dunn's indices, to relational data. In doing so, we develop a framework to convert many other validity indices to a relational form. Numerical examples on 12 datasets (samples from four normal mixtures, four real-world object datasets, and four real-world “pure relational” datasets) using the relational duals of the hard, fuzzy, and possibilistic c-means cluster algorithms are offered to illustrate and evaluate the new indices.

C*-algebras and self-similar groups

C*-algebras and self-similar groups

Groups generated by 3-state automata over a 2-letter alphabet. II

This is the second in a series of papers presenting results on the classification of groups generated by 3-state automata over a 2-letter alphabet. Among the examples treated here, one can find automata generating the free product of 3 cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups, and other interesting self-similar groups.

A correspondence between a class of monoids and self-similar group actions I

We describe a correspondence between a class of left cancellative monoids and self-similar group actions in the sense of Nekrashevych et al. This correspondence originated in Perrot’s 1972 thesis, and developed the ideas to be found in Rees’ 1948 paper.

Self-similar groups and their geometry

This is an overview of results concerning applications of self-similar groups generated by automata to fractal geometry and dynamical systems. Few proofs are given, interested reader can find the rest of the proofs in the monograph [Nek05]. We associate to every contracting self-similar action a topological space JG called limit space together with a surjective continuous map s : JG −→ JG. On the other hand if we have an expanding self-covering f : M1 −→ M of a topological space by its open subset, then we construct the iterated monodromy group (denoted IMG(f)) of f , which is a contracting self-similar group. These two constructions (dynamical system (JG, s) from a self-similar group and self-similar group IMG(f) from a dynamical system) are inverse to each other. The action of f on its Julia set is topologically conjugate to the action of s on the limit space JIMG(f) (see Theorem 6.1). We get in this way on one hand interesting examples of groups from dynamical systems (like the “basilica group” IMG ( z − 1 ) , which is a first example of an amenable group not belonging to the class of the sub-exponentially amenable groups). On the other hand, iterated monodromy groups are algebraic tools giving full information about combinatorics of self-coverings. The paper has the following structure. Section “Self-similar actions and automata” provides the basic notions from automata theory and theory of groups acting on rooted trees. It also gives some classical examples of self-similar groups. Section “Permutational bimodules” develops algebraic tools which are used in the study of self-similar groups. We define the notion of a permutational bimodule, which gives a convenient algebraic interpretation of automata. A closely related notion is virtual endomorphism, which can be used to construct explicit self-similar actions. We describe at the end of the section self-similar actions of free abelian groups and show how they are related to numeration systems on Z. Section 4 defines iterated monodromy groups. We show how to compute them (their standard actions) as groups generated by automata. Section 5 studies contracting self-similar actions and defines their limit spaces JG. We also prove some basic properties of the limit spaces, limit G-spaces and tiles. The last section shows connections of the obtained results with other topics of Mathematics. Subsection 6.2 shows that Julia sets of post-critically finite rational functions are limit spaces of their iterated monodromy groups. Next two subsections show a connection between topology of the limit spaces and a notion of bounded automata from [Sid00] and construct an iterative algorithm finding approximations of the limit space of actions by bounded automata. In Subsection 6.5 automata generating iterated monodromy groups of complex polynomials are described. We will see in particular, that iterated monodromy groups of complex polynomials are generated by bounded automata, so that the algorithm of the previous subsection can be used to draw topological approximations of the Julia sets of polynomials. We study in the last subsection the limit spaces of free Abelian groups and fit the theory of self-affine “digit” tilings in the framework of self-similar groups and their limit spaces.

Groups generated by 3-state automata over a 2-letter alphabet, I

This is the second in a series of papers presenting results on the classification of groups generated by 3-state automata over a 2-letter alphabet. Among the examples treated here, one can find automata generating the free product of 3 cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups, and other interesting self-similar groups.

Book Review: Self-similar groups

Book Review: Self-similar groups

Hausdorff dimension in a family of self-similar groups

For every prime p and every monic polynomial f, invertible over p, we define a group G p, f of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group \(G_{2, x^{2}+x+1}\). We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further “finiteness” properties are also discussed (amenability, torsion and intermediate growth).

Self-similar groups, operator algebras and Schur complement

In the first part of the article we introduce $C$*-algebras associated to self-similar groups and study their properties and relations to known algebras. The algebras are constructed as subalgebras of the Cuntz-Pimsner algebra (and its homomorphic images) associated with the self-similarity of the group. We study such properties as nuclearity, simplicity and Morita equivalence with algebras related to solenoids. &nbsp The second part deals with Schur complement transformations of elements of self-similar algebras. We study the properties of such transformations and apply them to the spectral problem for Markov type elements in self-similar $C$*-algebras. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. Application of the Schur complement method in many situations reduces the spectral problem to study of invariant sets (very often of the type of a strange attractor'') of a multidimensional rational transformation. A number of illustrating examples is provided. Finally, we observe a relation between Schur complement transformations and Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar groups.

"MÜNCHHAUSEN TRICK" AND AMENABILITY OF SELF-SIMILAR GROUPS

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μw. If the measure μ is self-similar in the sense that μw is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

A self-similar inhomogeneous dust cosmology

A detailed study of an inhomogeneous dust cosmology contained in a -law family of perfect-fluid metrics recently presented by Mars and Senovilla is performed. The metric is shown to be the most general orthogonally transitive, Abelian, on solution admitting an additional homothety such that the self-similar group is of Bianchi type VI and the fluid flow is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V, VIII and IX are shown to be impossible thus making this metric privileged from a mathematical viewpoint. The differential equations determining the metric are partially integrated and the line-element is given up to a first-order differential equation of Abel type of first kind and two quadratures. The solutions are qualitatively analysed by investigating the corresponding autonomous dynamical system. The spacetime is regular everywhere except for the big bang and the metric is complete both into the future and in all spatial directions. The energy density is positive, bounded from above at any instant of time and with an spatial profile (in the direction of inhomogeneity) which is oscillating with a rapidly decreasing amplitude. The generic asymptotic behaviour at spatial infinity is a homogeneous plane wave. Well known dynamical system results indicate that this metric is very likely to describe the asymptotic behaviour in time of a much more general class of inhomogeneous dust cosmologies.

Inhomogeneous self-similar cosmological models

A class of self-similar spatially inhomogeneous models whose time evolution is determined by the similarity group is investigated. The self-similarity group is three-dimensional and acts transitively on timelike hypersurfaces, reducing the Einstein equations to a system of ordinary differential equations describing the dependence of the metric variables on the remaining spatial coordinate. This system of ordinary differential equations is given a Hamiltonian formulation which then serves as a starting point for a search for 'hidden' symmetries. Several cases are shown to allow such symmetries. Exploiting these symmetries makes it possible to integrate the field equations and thereby obtain several new classes of inhomogeneous self-similar exact solutions.