Subgroups of unitriangular groups of infinite matrices

We show, that for any associative ring $R$, the subgroup $\mathrm{UT}_r(\infty,R)$ of row finite matrices in $\mathrm{UT}(\infty,R)$, the group of all infinite dimensional (indexed by $\mathbb N$) upper unitriangular matrices over $R$, is generated by strings (block-diagonal matrices with finite blocks along the main diagonal). This allows us to define a large family of subgroups of $\mathrm{UT}_r(\infty,R)$ associated to some growth functions. The smallest subgroup in this family, called the group of banded matrices, is generated by 1-banded simultaneous elementary transvections (a slight generalization of the usual notion of elementary transvections). We introduce a notion of net subgroups and characterize the normal net subgroups of $\mathrm{UT}(\infty,R)$.