The lower central series of subgroups of the Vershik–Kerov group

We describe the derived and the lower central series of certain subgroups of the Vershik–Kerov group. We are concerned with the subgroups consisting of infinite matrices having finite number of nonzero entries in each row. We consider the group of matrices over rings which are associative, commutative, of stable rank at most one and such that the identity can be written as a sum of two units. For this case we give a complete description of the derived and the lower central series. Moreover, we prove that every element of discussed commutator subgroups can be written as a product of a finite number of commutators.