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

This article is cited in 63 scientific papers (total in 64 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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    2. A. Ershler (Dyubina), “The asymptotics of drift and entropy for a random walk on groups”, Russian Math. Surveys, 56:3 (2001), 580–581  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Lindenstrauss E., Peres Y., Schlag W., “Bernoulli convolutions and an intermediate value theorem for entropies of $K$-partitions”, J. Anal. Math., 87:1 (2002), 337–367  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. Bufetov A.I., “Convergence of spherical averages for actions of free groups”, Ann. of Math. (2), 155:3 (2002), 929–944  crossref  mathscinet  zmath  isi  elib  scopus
    5. Nechaev S., Voituriez R., “Random walks on three-strand braids and on related hyperbolic groups”, J. Phys. A, 36:1 (2003), 43–66  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    6. A. V. Malyutin, “The Poisson–Furstenberg boundary of а locally free group”, J. Math. Sci. (N. Y.), 129:2 (2005), 3787–3795  mathnet  crossref  mathscinet  zmath  elib
    7. Revelle D., “Rate of escape of random walks on wreath products and related groups”, Ann. Probab., 31:4 (2003), 1917–1934  crossref  mathscinet  zmath  isi  elib  scopus
    8. Kapovich I., Miasnikov A., Schupp P., Shpilrain V., “Generic-case complexity, decision problems in group theory, and random walks”, J. Algebra, 264:2 (2003), 665–694  crossref  mathscinet  zmath  isi  elib  scopus
    9. Ageev O.N., “On asymmetry of the future and the past for limit self-joinings”, Proc. Amer. Math. Soc., 131:7 (2003), 2053–2062  crossref  mathscinet  zmath  isi  elib  scopus
    10. Kaimanovich V.A., “Random walks on Sierpinski hyperbolicity and Stochastic graphs: Homogenization”, Fractals in Graz 2001: Analysis - Dynamics - Geometry - Stochastics, Trends in Mathematics, 2003, 145–183  mathscinet  zmath  isi
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    14. Erschler A., “Boundary behavior for groups of subexponential growth”, Ann. of Math. (2), 160:3 (2004), 1183–1210  crossref  mathscinet  isi  elib  scopus
    15. Okayasu R., “Entropy of subshifts and the Macaev norm”, J. Math. Soc. Japan, 56:1 (2004), 177–191  crossref  mathscinet  zmath  isi  elib  scopus
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    23. Kapovich I., Schupp P., “Genericity, the Arzhantseva-Ol'shanskii method and the isomorphism problem for one-relator groups”, Math. Ann., 331:1 (2005), 1–19  crossref  mathscinet  zmath  isi  elib  scopus
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