Hyperbolic groups and their quotients of bounded exponents

Ivanov, S.; Ol’shanskii, A.

doi:10.1090/S0002-9947-96-01510-3

Hyperbolic groups and their quotients of bounded exponents
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by S. V. Ivanov and A. Yu. Ol’shanskii PDF

Trans. Amer. Math. Soc. 348 (1996), 2091-2138 Request permission

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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