On the width of verbal subgroups of the groups of triangular matrices over a field of arbitrary characteristic

The width [Formula: see text] of the verbal subgroup [Formula: see text] of a group [Formula: see text] defined by a collection of group words [Formula: see text] is the smallest number [Formula: see text] in [Formula: see text] such that every element of [Formula: see text] is the product of at most [Formula: see text] words in [Formula: see text] evaluated on [Formula: see text] and their inverses. Well known that every verbal subgroup of the group [Formula: see text] of triangular matrices over an arbitrary field [Formula: see text] can be defined by just one word: an outer commutator word or a power word. We prove that [Formula: see text] for every outer commutator word [Formula: see text] and that [Formula: see text] except for two cases, when it is equal to 2. For finitary triangular groups, the situation is similar.

A focal subgroup theorem for outer commutator words

Let

FREE GROUPS OF OUTER COMMUTATOR VARIETIES OF GROUPS

If F is a free group, 1 < i [les ] j [les ] 2i and i [les ] k [les ] i + j + 1 then F/[γj(F), γi(F), γk(F)] is residually nilpotent and torsion-free. This result is extended to 1 < i [les ] j [les ] 2i and i [les ] k [les ] 2i + 2j. It is proved that the analogous Lie rings, L/[Lj, Li, Lk] where L is a free Lie ring, are torsion-free. Candidates are found for torsion in L/[Lj, Li, Lk] whenever k is the least of {i, j, k}, and the existence of torsion in L/[Lj, Li, Lk] is proved when i, j, k [les ] 5 and k is the least of {i, j, k}.

On Outer-Commutator Words

Let F be the group freely generated by the countably infinite set X = {x1, x2, . . . ,xi, . . . }. Let w(x1, x2, . . . , xn) be a reduced word representing an element of F and let G be an arbitrary group. Then V(w, G) will denote the setwhose elements will be called values of w in G. The subgroup of G generated by V(w, G) will be called the verbal subgroup of G with respect to w and be denoted by w(G).