The local structure of groups of triangular automorphisms of relatively free algebras

Let $K$ be an arbitrary field and $C_n$ a relatively free algebra of rank $n$. In particular, as $C_n$ we may treat a polynomial algebra $P_n$, a free associative algebra $A_n$, or an absolutely free algebra $F_n$. For the algebras $C_n=P_n$, $A_n$, $F_n$, it is proved that every finitely generated subgroup $G$ of a group $TC_n$ of triangular automorphisms admits a faithful matrix representation over a field $K$; hence it is residually finite by Mal’tsev's theorem. For any algebra $C_n$, the triangular automorphism group $TC_n$ is locally soluble, while the unitriangular automorphism group $UC_n$ is locally nilpotent. Consequently, $UC_n$ is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup $G$ of $UC_n$ can be arbitrarily large with increasing $n$ or transcendence degree of a field $K$ over its prime subfield.