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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 \(\operatorname{Aut}(F_n)\).

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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