The local structure of groups of triangular automorphisms of relatively free algebras.
Let \(C_n\) be a relatively free algebra of rank \(n\) in a variety of algebras over an arbitrary field \(K\). In particular, \(C_n\) may be the polynomial algebra \(P_n\), the free associative algebra \(A_n\), or the absolutely free algebra \(F_n\). In the paper under review the author studies the local properties of the group \(UC_n\) of unitriangular automorphisms and the group \(TC_n\) of triangular automorphisms of \(C_n\), i.e., the properties of their finitely generated subgroups. When \(C_n=P_n,A_n,F_n\) the author establishes that every finitely generated subgroup \(G\) of \(TC_n\) has a faithful matrix representation over the field \(K\). Combined with the classical theorem of Mal'tsev that finitely generated matrix groups are residually finite, this implies that \(G\) is residually finite. As a consequence, for an arbitrary relatively free algebra \(C_n\) the group \(TC_n\) is locally solvable and the group \(UC_n\) is locally nilpotent; both groups are locally linear and locally residually finite. The solvability of \(UC_n\) and its nonlinearity (when the characteristic of \(K\) is 0 and \(n\geq 4\)) were established by \textit{V. A. Roman'kov, I. V. Chirkov} and \textit{M. A. Shevelin}, [Sib. Mat. Zh. 45, No. 5, 1184-1188 (2004); translation in Sib. Math. J. 45, No. 5, 974-977 (2004; Zbl 1059.17004)]. The nonlinearity for \(P_n\), \(n\geq 3\), over an arbitrary field \(K\) was given by \textit{Yu. V. Sosnovskij}, Sib. Mat. Zh. 48, No. 3, 689-693 (2007); translation in Sib. Math. J. 48, No. 3, 555-558 (2007; Zbl 1155.14315)]. The recent paper by \textit{V. G. Bardakov, M. V. Neshchadim} and \textit{Yu. V. Sosnovsky} [J. Algebra 362, 201-220 (2012; Zbl 1269.16033)] contains a detailed study of the groups of triangular automorphisms of \(P_n\) and \(A_n\). Finally, the present paper shows that the width of the commutator subgroup (the minimal number of commutators needed to write the elements of the commutator subgroup) of a finitely generated subgroup \(G\) of \(UC_n\) can be arbitrarily large when \(n\) and the transcendence degree of \(K\) over the prime subfield increase. It is interesting to compare this result with the result in the above cited paper by Bardakov, Neshchadim, and Sosnovsky that for unitary algebras and fields of characteristic 0 every element of the commutator subgroup of \(UP_n\) and \(UA_n\) can be written as a single commutator.
scientific article; zbMATH DE number 6138173