The linearity problem for the unitriangular automorphism groups of free groups

We prove that the unitriangular automorphism group of a free group of rank $n$ has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if $n\leq3$. Thus, we have completed a description of relatively free groups with linear the unitriangular automorphism groups. This description was initiated by Erofeev and the author in [1], where proper varieties of groups have been considered.