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 Tr. Mat. Inst. Steklova, 2011, Volume 273, Pages 72–191 (Mi tm3283)

Some topics in the dynamics of group actions on rooted trees

R. I. Grigorchukab

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
b Department of Mathematics, Texas A&M University, College Station, TX, USA

Abstract: 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 self-similar 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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 273, 64–175

Bibliographic databases:

UDC: 512+517.98+519.1

Citation: R. I. Grigorchuk, “Some topics in the dynamics of group actions on rooted trees”, Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Tr. Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 72–191; Proc. Steklov Inst. Math., 273 (2011), 64–175

Citation in format AMSBIB
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Related articles on Google Scholar: Russian articles, English articles

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