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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hyperbolic groups and their
quotients of bounded exponents


Authors: S. V. Ivanov and A. Yu. Ol'shanskii
Journal: Trans. Amer. Math. Soc. 348 (1996), 2091-2138
MSC (1991): Primary 20F05, 20F06, 20F32, 20F50
MathSciNet review: 1327257
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Abstract: In 1987, Gromov conjectured that for every non-elementary hyperbolic group $G$ there is an $n =n(G)$ such that the quotient group $G/G^{n}$ is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of $G/G^{n}$ is given, it is proven that the word and conjugacy problem are solvable in $G/G^{n}$ and that $\bigcap _{k=1}^{\infty }G^{k} = \{ 1\}$. The proofs heavily depend upon prior authors' results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents.


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Additional Information

S. V. Ivanov
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: ivanov@math.uiuc.edu

A. Yu. Ol'shanskii
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia
Email: olsh@nw.math.msu.su

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01510-3
PII: S 0002-9947(96)01510-3
Received by editor(s): April 5, 1995
Additional Notes: The second author was supported in part by Russian Fund for Fundamental Research, Grant 010-15-41, and by International Scientific Foundation, Grant MID 000.
Article copyright: © Copyright 1996 American Mathematical Society