We give examples to show how self-similar groups defined in this way can be separated into different tree language hierarchies.Source: “Proposals to Revise the Virginia Constitution: I. ![]() of finite type, sofic) as in symbolic dynamics. Rabin, Büchi, etc.), or considered as elements of a tree shift (e.g. The elements of such a group correspond to labeled trees which may be recognized by a tree automaton (e.g. We generalize the notion of self-similar groups of infinite tree automorphisms to allow for groups which are defined on a tree but may not act faithfully on it. Fi- nally we observe a relation between the Schur complement transformations and Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar groups. A number of illustrating examples is provided. 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. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. We study properties of such transformations and apply them to the spectral problem for Markov type elements in self- similar Calgebras. The second part deals with the Schur complement transformations of el- ements of self-similar algebras. We study such properties as nuclearity, simplicity and Morita equiva- lence with algebras related to solenoids. The algebras are constructed as sub-algebras of the Cuntz-Pimsner algebra (and its homomorphic images) associated with the self-similarity of the group. In the first part of the article we introduce C�-algebras associ- ated to self-similar groups and study their properties and relations to known algebras. An appendix briefly reviews the connection between Jacobi operators on simple Toeplitz subshifts and Laplacians on Schreier graphs of self-similar groups. This approach allows us to establish uniformity of cocycles for simple Toeplitz subshifts and Sturmian subshifts in a unified way. To do so, we use the so-called leading sequence condition for subshifts, which stems from a collaboration with Grigorchuk, Lenz and Nagnibeda, see. In fact we prove the stronger statement that every locally constant SL(2,R)-cocycle is uniform. In addition the spectrum is shown to be a Cantor set of Lebesgue measure zero. This generalises a result of Grigorchuk, Lenz and Nagnibeda from 2018. Regarding the Jacobi operators, we show that they have empty pure point spectrum for almost all elements in the subshift. These combinatorial results can also be found in. We characterise alpha-repetitivity and, based on a work by Liu and Qu from 2011, the Boshernitzan condition. In addition we give a complete description of the de Bruijn graphs. More precisely, we derive explicit formulas for complexity, palindrome complexity and, for sufficiently large word length, repetitivity. We investigate combinatorial properties of aperiodic simple Toeplitz subshifts, as well as spectral properties of Jacobi operators defined by them. Key words: minimal system of generators wreath product Sylow subgroups group semidirect product. The main result is the proof of minimality of this generating set of the above described subgroups and also the description of their structure. Also, the goal of this paper is to investigate the structure of 2-sylow subgroup of alternating group more exactly and deep than in articlle of U.~Dmitruk and V.~Suschansky. ![]() For the construction of minimal generating set we used the representation of elements of group by automorphisms of portraits for binary tree. In other words, the problem is not simply in the proof of existence of a generating set with elements for Sylow 2-subgroup of alternating group of degree $2^k$ and its constructive proof and proof its minimality. ![]() The aim of this paper is to research the structure of Sylow 2-subgroups and to construct a minimal generating system for such subgroups. The authors of article "Structure of 2-sylow subgroup of symmetric and alternating group" U.~Dmitruk, V.~Suschansky didn't proof minimality of finding by them system of generators for such Sylow 2-subgroups of $A_n$ and structure of it were founded only descriptively. This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ $ of alternating group, finding structure of these subgroups.
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