On the outer automorphism groups of free groups. (Q743907)

Let \(F_n\) be a free group of finite rank \(n\geq 3\). Firstly \textit{D. G. Khramtsov} [in: Teoretiko-gruppovye issledovaniya. Sbornik nauchnykh trudov. Sverdlovsk: Ural'skoe Otdelenie AN SSSR. 128-143 (1990; Zbl 0808.20036)] and later, with a different approach, \textit{M. R. Bridson} and \textit{K. Vogtmann}, [J. Algebra 229, No. 2, 785-792 (2000; Zbl 0959.20027)] proved that the outer automorphism group \(\mathrm{Out}(F_n)\) is complete. The completeness of \(\mathrm{Out}(F_n)\) is related with the completeness of the automorphism group \(\Aut(F_n)\). \textit{J. L. Dyer} and \textit{E. Formanek}, [J. Lond. Math. Soc., II. Ser. 11, 181-190 (1975; Zbl 0313.20021)], proved that \(\Aut(F_n)\) is complete for \(n\geq 2\). In the case where the free rank of a free group \(F\) is infinite the study of the completeness of \(\Aut(F)\) and of \(\mathrm{Out}(F)\) needs a different manipulation. In [J. Lond. Math. Soc., II. Ser. 61, No. 2, 423-440 (2000; Zbl 0963.20015)] the author proved that \(\Aut(F)\) is complete for any infinitely generated free group \(F\). Here the author proves that \(\mathrm{Out}(F)\) is complete for every countably generated free group \(F\). For a group \(G\) a relation \(R\subseteq G^n\) is called definable in \(G\) if it can be characterized in \(G\) in terms of the group operation. An involution \(\varphi\in\Aut(G)\), where \(G\) is a relatively free group, is said to be extremal if there is a basis \(\mathcal B\) of \(G\) such that \(\varphi\) inverts an element of \(\mathcal B\) and fixes all other elements of \(\mathcal B\). An involution \(f\in\mathrm{Out}(G)\) is said be extremal if it induced by an extremal involution of \(\Aut(G)\). The author studies the definability of certain families of involutions in \(\mathrm{Out}(F)\) and obtains an explicit group-theoretic characterization of the extremal involutions in \(\mathrm{Out}(F)\). This enables him to prove that an arbitrary automorphism \(\Delta\in\Aut(\mathrm{Out}(F))\) can be composed with a suitable inner automorphism to ensure that the resulting automorphism \(\Delta'\) fixes point-wise the images of all \(\mathcal B\)-finitary automorphisms of the group \(F\) in \(\mathrm{Out}(F)\). Finally it is proved that necessarily \(\Delta'\) is trivial, which completes the proof. In the last step the author makes use of the notion of the small index property. Let \(\mathcal F\) be a relatively free algebra of infinite rank \(\mathcal N\). It is said that \(\mathcal F\) has the small index property, if any subgroup \(\Sigma\) of the automorphism group \(\Aut(\mathcal F)\) of index at most \(\mathcal N\) contains the point-wise stabilizer \(\Gamma(U)\) of a subset \(U\) of \(\mathcal F\) of cardinality \(<\mathcal N\). In [\textit{R. M. Bryant} and \textit{D. M. Evans}, J. Lond. Math. Soc., II. Ser. 55, No. 2, 363-369 (1997; Zbl 0867.20032)] the small index is established only for the free groups of countably infinite rank. Therefore, as the author points out, it is not possible to apply similar arguments to prove (as he believes) that \(\mathrm{Out}(F)\) is complete for a free group \(F\) of arbitrary infinite rank.