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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
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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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http://mi.mathnet.ru/eng/mz4890https://doi.org/10.4213/mzm4890 http://mi.mathnet.ru/eng/mz/v85/i4/p516
Citing articles on Google Scholar:
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This publication is cited in the following articles:
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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
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V. S. Atabekyan, “Monomorphisms of Free Burnside Groups”, Math. Notes, 86:4 (2009), 457–462
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V. S. Atabekyan, “Nonunitarizable Periodic Groups”, Math. Notes, 87:6 (2010), 908–911
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S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855
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A. S. Pahlevanyan, “Independent pairs in free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 2, 58–62
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H. R. Rostami, “Non-unitarizable groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2010, no. 3, 40–43
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V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24
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Coulon R., “Growth of Periodic Quotients of Hyperbolic Groups”, Algebr. Geom. Topol., 13:6 (2013), 3111–3133
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S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
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S. I. Adian, V. S. Atabekyan, “$C^*$-Simplicity of $n$-Periodic Products”, Math. Notes, 99:5 (2016), 631–635
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Adian S.I. Atabekyan V.S., “Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117
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