Verbal subgroups in the group of triangular matrices over field of characteristic 0.
The author considers the verbal subgroups of the triangular group \(T=Tr(n,F)\) and the unitriangular group \(U=Tr_1(n,F)\), in both cases where \(n\) is a positive integer and \(F\) is a field of characteristic zero. She proves that the only verbal subgroups of \(U\) are the terms of its lower central series, while the verbal subgroups of \(T\) are either the terms of the lower central series of \(U\) or are of the form \(VU\), where \(V\) is a verbal subgroup of the full diagonal group \(D(n,F)\). These results are then extended to the direct limits over \(n\) of the \(Tr(n,F)\) and the \(Tr_1(n,F)\) with the obvious top left corner embeddings. Let \(c_k\) denote the commutator word \([x_1,x_2,\dots,x_k]\) defining the \(k\)-th term of the lower central series of a group. The author's final theorem is that every element of the \(k\)-th term of the lower central series of \(T\), \(U\) and their two direct limits over \(n\) (see above) is a value of \(c_k\) in the group.
scientific article; zbMATH DE number 5545844