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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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