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Rostislav Grigorchuk

Publications and source records attributed to Rostislav Grigorchuk.

62 records · Page 4Linked to original sources
arXiv2007

Schreier spectrum of the Hanoi Towers group on three pegs

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.

math.GR↗
arXiv2007

Homomorphic images of Branch groups, and Serre's property (FA)

It is shown that a finitely generated branch group has Serre's property (FA) if and only if it does not surject onto the infinite cyclic group or the infinite dihedral group. An example of a finitely generated self-similar branch group surjecting onto the infinite cyclic group is constructed.

math.GR↗
arXiv2007

The Spectral Problem, Substitutions and Iterated Monodromy

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.

math.GR↗
arXiv2007

Self-similar groups, algebras and Schur complements

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 sub-algebras 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. The second part deals with the Schur complement transformations of elements of self-similar algebras. We study 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 the Schur complement transformations and Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar groups.

math.GR↗
arXiv2006

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

An approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented. Several results related to the whole class are formulated. In particular, all finite, abelian, and free groups are classified. In addition, we provide detailed information and complete proofs for several groups from the class, with the intention of showing the main methods and techniques used in the classification.

math.GR↗
arXiv2006

Asymptotic aspects of Schreier graphs and Hanoi Towers groups

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.

math.GR↗