Verbal subgroups in the group of triangular matrices over field of characteristic 0

In this paper we describe the verbal subgroups of groups UTn(K) and Tn(K), which are the groups of, respectively, unitriangular and triangular matrices over field K of characteristic 0. We prove that every verbal subgroup of UTn(K) coincides with a term of the lower central series of the group and that every verbal subgroup V(Tn(K)) either is of the form V(K)⋅UTn(K) or coincides with a verbal subgroup of UTn(K). We also describe the width of these verbal subgroups and make a few remarks regarding the groups of infinite triangular matrices.