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Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 6, Pages 1139–1149 (Mi izv1699)  

This article is cited in 31 scientific papers (total in 32 papers)

Random walks on free periodic groups

S. I. Adian

Abstract: An upper estimate is obtained for the growth exponent of the set of all uncancellable words equal to $1$ in a group given by a system of defining relations with the Dehn condition. By a theorem of Grigorchuk, this yields a sufficient test for the transience of a random walk on a group given by a system of defining relations with the Dehn condition, and for the nonamenability of such a group. It is proved that the free periodic groups $\mathbf B(m,n)$ with $m\geqslant2$ and odd $n\geqslant665$ satisfy this test. A question asked by Kesten in 1959 is thereby answered in the negative, and a conjecture put foth earlier by the author is confirmed.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1983, 21:3, 425–434

Bibliographic databases:

UDC: 519.4
MSC: Primary 20E05, 20F50, 60J15; Secondary 20F05, 20FD6, 20F10, 20F19
Received: 08.06.1982

Citation: S. I. Adian, “Random walks on free periodic groups”, Izv. Akad. Nauk SSSR Ser. Mat., 46:6 (1982), 1139–1149; Math. USSR-Izv., 21:3 (1983), 425–434

Citation in format AMSBIB
\by S.~I.~Adian
\paper Random walks on free periodic groups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1982
\vol 46
\issue 6
\pages 1139--1149
\jour Math. USSR-Izv.
\yr 1983
\vol 21
\issue 3
\pages 425--434

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    This publication is cited in the following articles:
    1. R. I. Grigorchuk, “Degrees of growth of finitely generated groups, and the theory of invariant means”, Math. USSR-Izv., 25:2 (1985), 259–300  mathnet  crossref  mathscinet  zmath
    2. Francçoise Point, “Groups with identities”, Annals of Pure and Applied Logic, 45:2 (1989), 171  crossref
    3. V. D. Mazurov, “Solved problems in the Kourovka Notebook”, Russian Math. Surveys, 46:5 (1991), 137–182  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. R. Grigorchuk, P. Harpe, “On problems related to growth, entropy, and spectrum in group theory”, J Dyn Control Syst, 3:1 (1997), 51  crossref  mathscinet  zmath  elib
    5. T CECCHERINISILBERSTEIN, R GRIGORCHUK, P DELAHARPE, “Décompositions paradoxales des groupes”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 327:2 (1998), 127  crossref
    6. P. de la Harpe, R. I. Grigorchuk, T. Ceccherini-Silberstein, “Amenability and Paradoxical Decompositions for Pseudogroups and for Discrete Metric Spaces”, Proc. Steklov Inst. Math., 224 (1999), 57–97  mathnet  mathscinet  zmath
    7. S. I. Adian, “The Burnside Problem on Periodic Groups, and Related Problems”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S2–S12  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. V. S. GUBA, “ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F”, Int. J. Algebra Comput, 14:05n06 (2004), 677  crossref
    9. G.N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, E. Ventura, “Uniform non-amenability”, Advances in Mathematics, 197:2 (2005), 499  crossref
    10. L. D. Beklemishev, I. G. Lysenok, A. A. Mal'tsev, S. P. Novikov, M. R. Pentus, A. A. Razborov, A. L. Semenov, V. A. Uspenskii, “Sergei Ivanovich Adian (on his 75th birthday)”, Russian Math. Surveys, 61:3 (2006), 575–588  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Yves de Cornulier, Romain Tessera, “Quasi-isometrically embedded free sub-semigroups”, Geom Topol, 12:1 (2008), 461  crossref  mathscinet  zmath  isi  elib
    12. V. S. Atabekyan, A. S. Pailevanyan, “Vlozhenie absolyutno svobodnykh grupp v gruppy $B(m,n,1)$”, Uch. zapiski EGU, ser. Fizika i Matematika, 2008, no. 3, 25–33  mathnet
    13. V. S. Atabekyan, “Uniform Nonamenability of Subgroups of Free Burnside Groups of Odd Period”, Math. Notes, 85:4 (2009), 496–502  mathnet  crossref  crossref  mathscinet  zmath  isi
    14. V. S. Atabekian, “On subgroups of free Burnside groups of odd exponent $n\ge 1003$”, Izv. Math., 73:5 (2009), 861–892  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. V. S. Atabekyan, “Monomorphisms of Free Burnside Groups”, Math. Notes, 86:4 (2009), 457–462  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. V. S. Atabekyan, “The normalizers of free subgroups in free Burnside groups of odd period $n\ge1003$”, J. Math. Sci., 166:6 (2010), 691–703  mathnet  crossref  mathscinet  elib
    17. V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911  mathnet  crossref  crossref  mathscinet  isi
    18. S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62  mathnet
    20. H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43  mathnet
    21. V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24  mathnet  crossref  mathscinet  isi  elib  elib
    22. L.S.. Cirio, Alessandro D'Andrea, Claudia Pinzari, Stefano Rossi, “Connected components of compact matrix quantum groups and finiteness conditions”, Journal of Functional Analysis, 2014  crossref
    23. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    24. S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71  mathnet  crossref  crossref  isi  elib
    25. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    26. S. I. Adian, V. S. Atabekyan, “$C^*$-Simplicity of $n$-Periodic Products”, Math. Notes, 99:5 (2016), 631–635  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    27. V. G. Durnev, O. V. Zetkina, A. I. Zetkina, “Ob amenabelnykh podgruppakh $F$-grupp”, Chebyshevskii sb., 17:2 (2016), 128–136  mathnet  elib
    28. Adian S.I., Atabekyan V.S., “Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117  crossref  isi
    29. Atabekyan V.S., Gevorgyan A.L., Stepanyan Sh.A., “The Unique Trace Property of N-Periodic Product of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:4 (2017), 161–165  crossref  isi
    30. Nekrashevych V., “Palindromic Subshifts and Simple Periodic Groups of Intermediate Growth”, Ann. Math., 187:3 (2018), 667–719  crossref  isi
    31. Coulon R., Gruber D., “Small Cancellation Theory Over Burnside Groups”, Adv. Math., 353 (2019), 722–775  crossref  isi
    32. V. S. Atabekyan, “The set of $2$-genereted $C^*$-simple relatively free groups has the cardinality of the continuum”, Uch. zapiski EGU, ser. Fizika i Matematika, 54:2 (2020), 81–86  mathnet  crossref
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