The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential −∆ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

/pdf/an-overview-of-periodic-elliptic-operators-uqeh1n79ah.pdf

An overview of periodic elliptic operators

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 ∈ {0, 1, 2}ℕ.

Solved and Unsolved Problems Around One Group

Asymptotic aspects of Schreier graphs and Hanoi Towers groups

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.

/pdf/some-topics-in-the-dynamics-of-group-actions-on-rooted-trees-drnk39evrv.pdf

Some topics in the dynamics of group actions on rooted trees

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. 
The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential $-\Delta+V(x)$, but to a wide variety of elliptic periodic equations and systems, equations on graphs, $\overline{\partial}$-operator, and other operators on abelian coverings of compact bases. 
Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3.

On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3

/pdf/on-a-free-group-of-transformations-defined-by-an-automaton-4p7d9x3e4s.pdf

We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

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Automata over a binary alphabet generating free groups of even rank

We introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata. We prove that transformations defined by automata from the first series generate a free non-Abelian group of infinite rank, while automata from the second series give rise to the free product of infinitely many groups of order 2.

https://www.ems-ph.org/journals/show_pdf.php?issn=1661-7207&vol=4&iss=2&rank=7

On a series of finite automata defining free transformation groups

We introduce two series of finite automata starting from the so-called Aleshin and Bellaterra automata. We prove that each automaton in the first series defines a free non-Abelian group while each automaton in the second series defines the free product of groups of order 2. Furthermore, these properties are shared by disjoint unions of any number of distinct automata from either series.

Mariya Vorobets

Papers

On a free group of transformations defined by an automaton

On a free group of transformations defined by an automaton

Automata over a binary alphabet generating free groups of even rank

On a series of finite automata defining free transformation groups

On a series of finite automata defining free transformation groups