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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 4(334), Pages 59–128 (Mi umn314)  

This article is cited in 69 scientific papers (total in 70 papers)

Dynamic theory of growth in groups: Entropy, boundaries, examples

A. M. Vershik

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: This paper deals with the following topics.
1) Numerical invariants of countable groups (entropy, logarithmic volume, and drift); the fundamental inequality relating these invariants, and comparison of generating sets on the basis of this inequality; Monte Carlo generation of groups.
2) An ergodic method for constructing and studying the boundaries of random walks, the entropy of the boundary polymorphism, and their relationship to the fundamental inequality.
3) A geometric realization of free soluble groups, their boundaries, and a geometric approach to the construction of normal forms in groups.
4) Local and locally free groups and calculation of constants for these groups.
5) Entropy in measure theory and in the theory of dynamical systems; new notions of entropy of a decreasing sequence of measurable partitions and secondary entropy of $K$-automorphisms.

DOI: https://doi.org/10.4213/rm314

Full text: PDF file (572 kB)
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English version:
Russian Mathematical Surveys, 2000, 55:4, 667–733

Bibliographic databases:

UDC: 512.54+519.21
MSC: Primary 37A35, 60G50, 28D20; Secondary 60B15, 20F10, 20F36, 20F16
Received: 09.03.2000

Citation: A. M. Vershik, “Dynamic theory of growth in groups: Entropy, boundaries, examples”, Uspekhi Mat. Nauk, 55:4(334) (2000), 59–128; Russian Math. Surveys, 55:4 (2000), 667–733

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