The non-finite presentability of \(IA(F_ 3)\) and \(\text{GL}_ 2(Z[t,t^{-1}])\)
If \(G'\) is the derived subgroup of the group \(G\), then every automorphism of \(G\) induces an automorphism on \(G/G'\) and the kernel of the relative homomorphism of \(\Aut G\) into \(\Aut(G/G')\) consists of all the automorphisms of \(G\), which induce the identity on \(G/G'\) and it is denoted by \(IA(G)\). In the case where \(G=F_n\) is the free group of rank \(n\), W. Magnus has shown that \(IA(F_n)\) is finitely generated and has exhibited a finite set of generators. But the question remained open whether the group \(IA(F_n)\) for \(n\geq 3\) is finitely presented. (For \(n=2\) it was proved by Nielsen that \(IA(F_2)\) is the group of inner automorphisms and therefore free.) In this paper the authors prove that the group \(IA(F_3)\) is not finitely presentable. In fact they prove something more general, i.e. that certain subgroups of \(IA(G)\), for \(G=F_3/N\), where \(N\) is a characteristic subgroup of \(F_3\), contained in the second derived subgroup \(F_3''\) of \(F_3\), is not finitely presentable. Their second main result is the proof that \(\text{GL}_2(\mathbb{Z}[t,t^{-1}])\) is not finitely presentable. Here also they prove something more. Namely, if \(R\) is a commutative ring and \(G\) an indicable group, then any subgroup \(H\) of \(\text{GL}_2(RG)\) which contains \(E_2(RG)\), is not finitely presentable. Here \(RG\) is the group ring of \(G\) over \(R\) and \(E_2(RG)\) is the elementary group of degree 2 over \(RG\).
scientific article; zbMATH DE number 1054511
scientific article