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Outer automorphisms of free Burnside groups. (English) Zbl 1291.20035
The main result of this paper is: Let \(1\to H\to G\to F\to 1\) be a short exact sequence of groups. Assume that \(H\) is non-trivial, finitely generated, \(G\) is hyperbolic, non-elementary, torsion-free and \(F\) is torsion-free. Then there exists an integer \(n_0\) such that, for all odd integers \(n\geq n_0\), the canonical map \(F\to\mathrm{Out}(H)\) induces an injective homomorphism \(F\hookrightarrow\mathrm{Out}(H/H^n)\).
Let \(\mathbf B_r(n)\) be the free Burnside group of rank \(r\) and exponent \(n\) and let \(\mathbf F_2\) be the free group of rank \(2\). As a consequence of the previous result the author obtains the following theorems.
Theorem 2. Let \(r\geq 3\), then there is an integer \(n_0\) such that for all odd integers \(n\geq n_0\), the group \(\mathrm{Out}(\mathbf B_r(n))\) contains a subgroup isomorphic to \(\mathbf F_2\).
Theorem 3. Let \(r\geq 1\), then there is an integer \(n_0\) such that for all odd integers \(n\geq n_0\), the groups \(\mathrm{Out}(\mathbf B_{2r}(n))\) and \(\mathrm{Out}(\mathbf B_{2r+1}(n))\) contain a subgroup isomorphic to \(\mathbb Z^r\).

MSC:
20F50 Periodic groups; locally finite groups
20E36 Automorphisms of infinite groups
20F28 Automorphism groups of groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
20E07 Subgroup theorems; subgroup growth
20F65 Geometric group theory
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