Large groups and their periodic quotients
Authors:
A. Yu. Olshanskii and D. V. Osin
Journal:
Proc. Amer. Math. Soc. 136 (2008), 753759
MSC (2000):
Primary 20F50, 20F05, 20E26
Published electronically:
November 26, 2007
MathSciNet review:
2361846
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We first give a short group theoretic proof of the following result of Lackenby. If is a large group, is a finite index subgroup of admitting an epimorphism onto a noncyclic free group, and are elements of , then the quotient of by the normal subgroup generated by is large for all but finitely many . In the second part of this note we use similar methods to show that for every infinite sequence of primes , there exists an infinite finitely generated periodic group with descending normal series , such that and is either trivial or abelian of exponent .
 1.
S.
V. Alešin, Finite automata and the Burnside problem for
periodic groups, Mat. Zametki 11 (1972),
319–328 (Russian). MR 0301107
(46 #265)
 2.
Benjamin
Baumslag and Stephen
J. Pride, Groups with two more generators than relators, J.
London Math. Soc. (2) 17 (1978), no. 3,
425–426. MR 0491967
(58 #11137)
 3.
E.
S. Golod, On nilalgebras and finitely approximable
𝑝groups, Izv. Akad. Nauk SSSR Ser. Mat. 28
(1964), 273–276 (Russian). MR 0161878
(28 #5082)
 4.
R.
I. Grigorchuk, Degrees of growth of finitely generated groups and
the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat.
48 (1984), no. 5, 939–985 (Russian). MR 764305
(86h:20041)
 5.
M.
Gromov, Hyperbolic groups, Essays in group theory, Math. Sci.
Res. Inst. Publ., vol. 8, Springer, New York, 1987,
pp. 75–263. MR 919829
(89e:20070), http://dx.doi.org/10.1007/9781461395867_3
 6.
Narain
Gupta and Saïd
Sidki, On the Burnside problem for periodic groups, Math. Z.
182 (1983), no. 3, 385–388. MR 696534
(84g:20075), http://dx.doi.org/10.1007/BF01179757
 7.
M. Lackenby, Adding high powered relations to large groups, prep., 2005; arXiv: math.GR/0512356.
 8.
Wilhelm
Magnus, Abraham
Karrass, and Donald
Solitar, Combinatorial group theory, 2nd ed., Dover
Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of
generators and relations. MR 2109550
(2005h:20052)
 9.
Hanna
Neumann, Varieties of groups, SpringerVerlag New York, Inc.,
New York, 1967. MR 0215899
(35 #6734)
 10.
A.
Yu. Ol′shanskiĭ, On residualing homomorphisms and
𝐺subgroups of hyperbolic groups, Internat. J. Algebra Comput.
3 (1993), no. 4, 365–409. MR 1250244
(94i:20069), http://dx.doi.org/10.1142/S0218196793000251
 11.
A.
Yu. Ol′shanskiĭ, On the BassLubotzky question about
quotients of hyperbolic groups, J. Algebra 226
(2000), no. 2, 807–817. MR 1752761
(2001i:20069), http://dx.doi.org/10.1006/jabr.1999.8170
 12.
V.
Ī. Suščans′kiĭ, Periodic
𝑝groups of permutations and the unrestricted Burnside
problem, Dokl. Akad. Nauk SSSR 247 (1979),
no. 3, 557–561 (Russian). MR 545692
(81k:20009)
 1.
 S. V. Aleshin, Finite automata and Burnside's problem on periodic groups, Math. Notes 11 (1972), no. 3, 199203. MR 0301107 (46:265)
 2.
 B. Baumslag and S. J. Pride, Groups with two more generators than relators, J. London Math. Soc. (2) 17 (1978), no. 3, 425426. MR 0491967 (58:11137)
 3.
 E. S. Golod, On nilalgebras and finitely approximable groups (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 273276. MR 0161878 (28:5082)
 4.
 R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSRIzv. 25 (1985), no. 2, 259300. MR 764305 (86h:20041)
 5.
 M. Gromov, Hyperbolic groups, Essays in Group Theory, MSRI Series, Vol. 8, (S. M. Gersten, ed.), Springer, 1987, 75263. MR 919829 (89e:20070)
 6.
 N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), 385386. MR 696534 (84g:20075)
 7.
 M. Lackenby, Adding high powered relations to large groups, prep., 2005; arXiv: math.GR/0512356.
 8.
 W. Magnus, A. Karras, and D. Solitar, Combinatorial group theory, Interscience Publ., 1966. MR 2109550 (2005h:20052)
 9.
 H. Neumann, Varieties of groups. SpringerVerlag New York, Inc., New York, 1967. MR 0215899 (35:6734)
 10.
 A. Yu. Olshanskii, On residualing homomorphisms and subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365409. MR 1250244 (94i:20069)
 11.
 A. Yu. Olshanskii, On the BassLubotzky question about quotients of hyperbolic groups, J. Algebra 226 (2000), no. 2, 807817. MR 1752761 (2001i:20069)
 12.
 V. I. Suschansky, Periodic groups of permutations and the general Burnside problem, Dokl. Akad. Nauk SSSR 247 (1979), no. 3, 557561. MR 545692 (81k:20009)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
20F50,
20F05,
20E26
Retrieve articles in all journals
with MSC (2000):
20F50,
20F05,
20E26
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/S0002993907091502
PII:
S 00029939(07)091502
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 050100895.
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.
