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Mat. Zametki, 2009, Volume 85, Issue 4, Pages 516–523 (Mi mz4890)  

This article is cited in 11 scientific papers (total in 11 papers)

Uniform Nonamenability of Subgroups of Free Burnside Groups of Odd Period

V. S. Atabekyan

Yerevan State University

Abstract: A famous theorem of Adyan states that, for any $m\ge 2$ and any odd $n\ge 665$, the free $m$-generated Burnside group $B(m,n)$ of period $n$ is not amenable. It is proved in the present paper that every noncyclic subgroup of the free Burnside group $B(m,n)$ of odd period $n\ge 1003$ is a uniformly nonamenable group. This result implies the affirmative answer, for odd $n\ge 1003$, to the following de la Harpe question: Is it true that the infinite free Burnside group $B(m,n)$ has uniform exponential growth? It is also proved that every $S$-ball of radius $(400n)^3$ contains two elements which form a basis of a free periodic subgroup of rank 2 in $B(m,n)$, where $S$ is an arbitrary set of elements generating a noncyclic subgroup of $B(m,n)$.

Keywords: free Burnside group, periodic group, amenable group, uniformly nonamenable groups, Følner constant, uniform exponential growth, hyperbolic group

DOI: https://doi.org/10.4213/mzm4890

Full text: PDF file (443 kB)
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English version:
Mathematical Notes, 2009, 85:4, 496–502

Bibliographic databases:

UDC: 512.543
Received: 22.04.2008
Revised: 30.06.2008

Citation: V. S. Atabekyan, “Uniform Nonamenability of Subgroups of Free Burnside Groups of Odd Period”, Mat. Zametki, 85:4 (2009), 516–523; Math. Notes, 85:4 (2009), 496–502

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

    This publication is cited in the following articles:
    1. 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
    2. V. S. Atabekyan, “Monomorphisms of Free Burnside Groups”, Math. Notes, 86:4 (2009), 457–462  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911  mathnet  crossref  crossref  mathscinet  isi
    4. 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
    5. A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62  mathnet
    6. H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43  mathnet
    7. 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
    8. Coulon R., “Growth of Periodic Quotients of Hyperbolic Groups”, Algebr. Geom. Topol., 13:6 (2013), 3111–3133  crossref  mathscinet  zmath  isi  elib  scopus
    9. 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
    10. 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
    11. Adian S.I. Atabekyan V.S., “Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117  crossref  mathscinet  zmath  isi  scopus
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