Subgroups of unitriangular groups of infinite matrices

We show that for any associative ring R, the subgroup UT _r(∞, R) of row-finite matrices in UT(∞, R), the group of all infinite-dimensional (indexed by ℕ) upper unitriangular matrices over R, is generated by the so-called strings (block-diagonal matrices with finite blocks along the main diagonal). This allows us to define a large family of subgroups of UT _r(∞, R) associated with 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 transvection). We introduce the notion of net subgroup and characterize the normal net subgroups of UT(∞, R). Bibliography: 26 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 137–154.