Related papers: On an uncountable family of graphs whose spectrum …

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…

With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{\Gamma_n\}_{n\geq 1}$, where $\Gamma_n$ is the Schreier graph…

Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the space of…

In this paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two…

This paper explores a previously uncharted automaton group generated by a 5-state automaton $(\Pi, A)$ acts by self-similarity on the regular rooted tree $A^{*}$ over a 2-letter alphabet set $A$. Group $G$ has been subjected to several…

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…

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of…

There is a recently discovered connection between the spectral theory of Schr\"o-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's…

Every self-similar group acts on the space $X^\omega$ of infinite words over some alphabet $X$. We study the Schreier graphs $\Gamma_w$ for $w\in X^\omega$ of the action of self-similar groups generated by bounded automata on the space…

Every automaton group naturally acts on the space $X^\omega$ of infinite sequences over some alphabet $X$. For every $w\in X^\omega$ we consider the Schreier graph $\Gamma_w$ of the action of the group on the orbit of $w$. We prove that for…

We study Schreier dynamical systems associated with a vast family of groups that hosts many known examples of groups of intermediate growth. We are interested in the orbital graphs for the actions of these groups on $d-$regular rooted trees…

We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and L functions of Schreier graphs of few groups namely the Grigorchuk group, Gupta-Sidki p group,…

Let $\Gamma$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \Gamma$, a domatic $\aleph_0$-partition (for its Schreier graph on $\Gamma$) is a partial function…

The present paper investigates the limit $G$-space $\mathcal{J}_{G}$ generated by the self-similar action of automatic groups on a regular rooted tree. The limit space $\mathcal{J}_{G}$ is the Gromov-Hausdorff limit of the family of…

We introduce the notion of the Automatic Logarithm $\mathcal L_{\mathcal A, \mathcal B}$ with the purpose of studying the expanding properties of Schreier graphs of action of the group generated by two finite initial Mealy automata…

In this paper, we determine precise formulas for the diameters, the number of perfect matchings, and the Tutte polynomials for an infinite family of finite graphs, namely the Schreier graphs of tree automaton groups, also called tree graph…

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This…

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the…

Let the group $G$ act transitively on the finite set $\Omega$, and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $Sch(G \circlearrowleft \Omega,S)$ is the graph with vertex set $\Omega$ and edge set $\{…

The $H$-join of a family of graphs $\mathcal{G}=\{G_1, \dots, G_p\}$, also called the generalized composition, $H[G_1, \dots, G_p]$, where all graphs are undirected, simple and finite, is the graph obtained by replacing each vertex $i$ of…