Free-by-finite cyclic automorphism groups (Q1801939)

This paper is a contribution to the search of finding groups which can be or cannot be automorphism groups of other groups. Motivated by a result of J. L. Dyer that the group \(SL_ 2(Z)\) is not the automorphism group of any finitely generated group, he determines the structure of the group \(G\) whose automorphism group \(A\) is isomorphic to the fundamental group of a graph of locally cyclic groups. He, roughly, proves that \(G\) is infinitely generated abelian and \(A\) is free-by-finite cyclic with all torsion elements having order dividing 8, 10, 12 or 30. One must note the corollaries: (1) With the exception of the finite cyclic and infinite dihedral groups, the fundamental group of a graph of locally cyclic groups is not the automorphism group of any finitely generated group. (2) Except for the infinite dihedral group, a nontrivial free-by-finite cyclic group whose center contains no element of order 2 is not the automorphism group of any group. (3) An infinite free-by-finite cyclic group which contains an element of finite order not dividing 8, 10, 12 or 30 is not the automorphism group of any group. For the proof the author uses the Bass-Serre theory of groups acting on trees.