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Hyperbolic groups and their quotients of bounded exponents. (English) Zbl 0876.20023

M. Gromov conjectured that for every non-elementary (that is, with no cyclic subgroup of finite index) hyperbolic group \(G\) there is an integer \(n=n(G)\) such that the factor group \(G/G^n\) is infinite, where \(G^n\) denotes the subgroup generated by all \(n\)th powers of the elements of \(G\). (In the special case of \(G\) being a non-cyclic free group such numbers \(n\) exist by the Adyan-Novikov negative solution of the Burnside Problem.)
The authors confirm this conjecture and prove that, moreover, there is \(n=n(G)\) such that (a) \(G/G^n\) is infinite; (b) the word and conjugacy problems are solvable in \(G/G^n\); (c) if \(n=n_1n_2\), where \(n_1\) is odd and \(n_2\) is a power of 2, then every finite subgroup of \(G/G^n\) is isomorphic to an extension of a finite subgroup of \(G\) by a subgroup of the direct product of two groups one of which is a dihedral group of order \(2n_1\) and the other is the direct product of several copies of a dihedral group of order \(2n_2\); (d) the subgroup \(G^n\) is torsion-free and \(\bigcap^\infty_{k=1}G^{kn}=1\).
The proof heavily depends on earlier results of the authors on Gromov’s conjecture for torsion-free hyperbolic groups and on the Burnside problem for groups of even exponent. Several open problems are stated, some of them looking to be of intermediate nature between the Burnside Problem and the Restricted Burnside Problem. Another interesting problem: is there a finitely presented group \(G\) such that \(\bigcap^\infty_{k=1}G^k=1\) and \(G\) is not residually finite?

MSC:

20F50 Periodic groups; locally finite groups
20F65 Geometric group theory
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
Full Text: DOI

References:

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