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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Large groups and their periodic quotients


Authors: A. Yu. Olshanskii and D. V. Osin
Journal: Proc. Amer. Math. Soc. 136 (2008), 753-759
MSC (2000): Primary 20F50, 20F05, 20E26
Published electronically: November 26, 2007
MathSciNet review: 2361846
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Abstract: We first give a short group theoretic proof of the following result of Lackenby. If $ G$ is a large group, $ H$ is a finite index subgroup of $ G$ admitting an epimorphism onto a non-cyclic free group, and $ g_1, \ldots, g_k$ are elements of $ H$, then the quotient of $ G$ by the normal subgroup generated by $ g_1^n, \ldots , g_k^n$ is large for all but finitely many $ n\in \mathbb{Z}$. In the second part of this note we use similar methods to show that for every infinite sequence of primes $ (p_1, p_2, \ldots )$, there exists an infinite finitely generated periodic group $ Q$ with descending normal series $ Q=Q_0\rhd Q_1\rhd \ldots $, such that $ \bigcap_i Q_i=\{ 1\} $ and $ Q_{i-1}/Q_i$ is either trivial or abelian of exponent $ p_i$.


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Additional Information

A. Yu. Olshanskii
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240; and Department of Mathematics, Moscow State University, Moscow, 119899, Russia
Email: alexander.olshanskiy@vanderbilt.edu

D. V. Osin
Affiliation: Department of Mathematics, The City College of New York, New York, New York 10031
Email: denis.osin@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09150-2
PII: S 0002-9939(07)09150-2
Received by editor(s): April 19, 2006
Published electronically: November 26, 2007
Additional Notes: The first author was supported in part by the NSF grants DMS 0245600 and DMS 0455881.
The second author was supported in part by NSF grant DMS 0605093. Both authors were supported in part by the Russian Fund for Basic Research grant 05-01-00895.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.