Self-similar Groups Research Articles

The M-polynomial of Schreier Graphs of the Basilica and Grigorchuk Groups: Comparative Evaluation

A focus for comprehending the complex structures present in self-similar groups has been the study of Schreier graphs related to the Basilica and Grigorchuk group in recent years. Basic to group theory, Schreier graphs give a geometric picture of these groups’ actions on sets and shed light on the connectivity and symmetry characteristics of these groups. This paper examines the M-polynomial of Schreier graphs of the Basilica and Grigorchuk groups, examining its consequences for topological indices and comparing the determined values.

𝐾𝐾-duality for self-similar groupoid actions on graphs

We extend Nekrashevych’s K K KK -duality for C ∗ C^* -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid ( G , E ) (G,E) acting faithfully on a finite directed graph E E , we associate two C ∗ C^* -algebras, O ( G , E ) \mathcal {O}(G,E) and O ^ ( G , E ) \widehat {\mathcal {O}}(G,E) , to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in K K KK -theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

Non-Hausdorff germinal groupoids for actions of countable groups

We study conditions under which the germinal groupoid associated to a minimal equicontinuous action of a countable group on a Cantor set has non-Hausdorff topology. We develop a new criterion, which serves as an obstruction for the étale topology on the groupoid to be non-Hausdorff. We use this and other criteria to study the topology of germinal groupoids for a few classes of actions. In particular, we give examples of families of contracting and non-contracting self-similar groups, which are amenable and whose actions have associated germinal groupoids with non-Hausdorff topology.

Iterated monodromy groups of exponential maps

This paper introduces iterated monodromy groups for transcendental functions and discusses them in the simplest setting, for post-singularly finite exponential functions. These groups are self-similar groups in a natural way, based on an explicit construction in terms of kneading sequences. We investigate the group theoretic properties of these groups, and show in particular that they are amenable, but they are not elementary subexponentially amenable.

Tree languages and branched groups

We study the portraits of isometries of rooted trees—the labelling of the tree, at each vertex, by the permutation of its descendants—in terms of languages. We characterize regularly branched self-similar groups in terms of omega -regular languages. We deduce the algorithmic decidability of some problems, such as the comparison of regularly branched contracting groups, and their orbit structure on the boundary of the rooted tree.

Braiding groups of automorphisms and almost-automorphisms of trees

AbstractWe introduce “braided” versions of self-similar groups and Röver–Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call “self-identical.” In particular, we use a braided version of the Grigorchuk group to construct a new group called the “braided Röver group,” which we prove is of type $\operatorname {\mathrm {F}}_\infty $ . Our techniques involve using so-called d-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

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.

Monadic second-order logic and the domino problem on self-similar graphs

We consider the domino problem on Schreier graphs of self-similar groups, and more generally their monadic second-order logic. On the one hand, we prove that if the group is bounded, then the domino problem on the graph is decidable; furthermore, under an ultimate periodicity condition, all its monadic second-order logic is decidable. This covers, for example, the Sierpiński gasket graphs and the Schreier graphs of the Basilica group. On the other hand, we prove undecidability of the domino problem for a class of self-similar groups, answering a question by Barbieri and Sablik, and study some examples including one of linear growth.

On the simplicity of Nekrashevych algebras of contracting self-similar groups

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych \(C^*\)-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the \(C^*\)-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta–Sidki groups, GGS-groups, multi-edge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk–Erschler group and Fabrykowski–Gupta group) and self-replicating spinal automaton groups.

The homology of the groupoid of the self-similar infinite dihedral group

We compute the $K$-theory of the $C^*$-algebra associated to the self-similar infinite dihedral group, and the homology of its associated étale groupoid. We see that the rational homology differs from the $K$-theory, strongly contradicting a conjecture posted by Matui. Moreover, we compute the abelianization of the topological full group of the groupoid associated to the self-similar infinite dihedral group.

Self-similarity and spectral dynamics

This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.

The Thue-Morse substitutions and self-similar groups and algebras

We introduce self-similar algebras and groups closely related to the Thue-Morse sequence, and begin their investigation by describing a~character on them, the `spread' character.

Extensions of automorphisms of self-similar groups

Abstract In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group L k = Z k ≀ Z \mathcal{L}_{k}=\mathbb{Z}_{k}\wr\mathbb{Z} .

Integrable and Chaotic Systems Associated with Fractal Groups.

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

Continuous orbit equivalences on self-similar groups

Continuous orbit equivalences on self-similar groups

Simplicity of inverse semigroup and étale groupoid algebras

In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an étale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential C⁎-algebras.The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies.Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic 0. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.

Generalized Grigorchuk’s overgroups as points in the space of marked 8-generated groups

First Grigorchuk group [Formula: see text] and Grigorchuk’s overgroup [Formula: see text], introduced in 1980, are self-similar branch groups with intermediate growth. In 1984, [Formula: see text] was used to construct the family of generalized Grigorchuk groups [Formula: see text], which has many remarkable properties. Following this construction, we generalize the Grigorchuk’s overgroup [Formula: see text] to the family [Formula: see text] of generalized Grigorchuk’s overgroups. We consider these groups as 8-generated and describe the closure of this family in the space [Formula: see text] of marked [Formula: see text]-generated groups.

Intransitive self-similar groups

A group is said to be self-similar provided it admits a faithful state-closed representation on some regular m-tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level of the tree. A standard approach for constructing a transitive self-similar representation of a group has been by way of a single virtual endomorphism of the group in question. Recently, it was shown that this approach when applied to the restricted wreath product Z≀Z could not produce a faithful transitive self-similar representations for any m≥2 (see [5]). In this work we study state-closed representations without assuming the transitivity condition. This general action is translated into a set of virtual endomorphisms corresponding to the different orbits of the action on the first level of the tree. In this manner, we produce faithful self-similar representations, some of which are also finite-state, for a number of groups such as Zω, Z≀Z and (Z≀Z)≀C2.

Spectral–Spatial Joint Sparse NMF for Hyperspectral Unmixing

The nonnegative matrix factorization (NMF) combining with spatial–spectral contextual information is an important technique for extracting endmembers and abundances of hyperspectral image (HSI). Most methods constrain unmixing by the local spatial position relationship of pixels or search spectral correlation globally by treating pixels as an independent point in HSI. Unfortunately, they ignore the complex distribution of substance and rich contextual information, which makes them effective in limited cases. In this article, we propose a novel unmixing method via two types of self-similarity to constrain sparse NMF. First, we explore the spatial similarity patch structure of data on the whole image to construct the spatial global self-similarity group between pixels. And according to the regional continuity of the feature distribution, the spectral local self-similarity group of pixels is created inside the superpixel. Then based on the sparse expression of the pixel in the subspace, we sparsely encode the pixels in the same spatial group and spectral group respectively. Finally, the abundance of pixels within each group is forced to be similar to constrain the NMF unmixing framework. Experiments on synthetic and real data fully demonstrate the superiority of our method over other existing methods.

Criticality in the conformational phase transition among self-similar groups in intrinsically disordered proteins: Probed by salt-bridge dynamics

Intrinsically disordered proteins (IDP) serve as one of the key components in the global proteome. In contrast to globular proteins, they harbor an enormous amount of physical flexibility enforcing them to be retained in conformational ensembles rather than stable folds. Previous studies in an aligned direction have revealed the importance of transient dynamical phenomena like that of salt-bridge formation in IDPs to support their physical flexibility and have further highlighted their functional relevance. For this characteristic flexibility, IDPs remain amenable and accessible to different ordered binding partners, supporting their potential multi-functionality. The current study further addresses this complex structure-functional interplay in IDPs using phase transition dynamics to conceptualize the underlying (avalanche type) mechanism of their being distributed across and hopping around degenerate structural states (conformational ensembles). For this purpose, extensive molecular dynamics simulations have been done and the data analyzed from a statistical physics perspective. Investigation of the plausible scope of 'self-organized criticality' (SOC) to fit into the complex dynamics of IDPs was found to be assertive, relating the conformational degeneracy of these proteins to their functional multiplicity. In accordance with the transient nature of 'salt-bridge dynamics', the study further uses it as a probe to explain the structural basis of the proposed criticality in the conformational phase transition among self-similar groups in IDPs. The analysis reveal scale-invariant self-similar fractal geometries in the structural conformations of different IDPs. The insights from the study has the potential to be extended further to benefit structural tinkering of IDPs in their functional characterization and drugging.