Every group is an outer automorphism group of a finitely generated group.
Due to the restriction to finitely generated groups, the title of the paper is slightly misleading: To be correct, it should begin with `Every countable group \dots'. Without the restriction to finitely generated groups it can be shown that any group is the outer automorphism group of some group. In fact the underlying groups can come from particular classes of groups as shown in recent papers and discussed in the introduction of this paper. An interesting application of small cancellation theory provides for every countable group \(Q\) the existence of a special finitely generated group \(N\) with \(\text{Out}(N)\cong Q\). The authors use a variant of \textit{E. Rips}' construction [Bull. Lond. Math. Soc. 14, 45-47 (1982; Zbl 0481.20020)].
scientific article; zbMATH DE number 2182121
scientific article