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Solvable subgroups of $$\text{Out}(F_n)$$ are virtually Abelian. (English) Zbl 1052.20027
It serves as a general guiding principle for the outer automorphism group $$\text{Out}(F_n)$$ of a free group $$F_n$$ that it behaves sometimes like the mapping class group of a compact surface (which is contained in $$\text{Out}(F_n)$$ if the surface is bounded, with free fundamental group of rank $$n$$), and sometimes like a linear group $$\text{GL}(n,\mathbb{Z})$$ (onto which $$\text{Out}(F_n)$$ projects by Abelianization). In a previous paper, the authors showed that $$\text{Out}(F_n)$$ satisfies the Tits alternative for linear groups, i.e. that subgroups are either virtually solvable or contain a free group of rank 2. The main result of the present paper states that every solvable subgroup of $$\text{Out}(F_n)$$ has a finite index subgroup that is finitely generated and free Abelian (for solvable subgroups of mapping class groups this is a result of J. S. Birman, A. Lubotzky and J. McCarthy [Duke Math. J. 50, 1107-1120 (1983; Zbl 0551.57004)], on the other hand, this is not true for linear groups noting that the Heisenberg group in $$\text{GL}(3,\mathbb{Z})$$ is solvable but not virtually Abelian). It is also shown that every Abelian subgroup of $$\text{Out}(F_n)$$ has a finite index subgroup that lifts to the automorphism group $$\operatorname{Aut}(F_n)$$.

##### MSC:
 20F28 Automorphism groups of groups 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 20E07 Subgroup theorems; subgroup growth 20F65 Geometric group theory 57M07 Topological methods in group theory
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