Free and Non-Free Subgroups of the Group UT(∞, ℤ)

We provide a method to find free groups of rank two in the group of infinite unitriangular matrices. Our groups are generated by two block-diagonal matrices, namely of the form A = diag(C, C, C...), B = diag(I_t, C, C...), where C is a matrix of finite dimension.We give a necessary and sufficient condition for A and B defined above to generate a free group when C is a transvection. We formulate a sufficient condition to generate a free group, when C is a product of any number of commuting transvections.We provide a classification of groups defined above, when C is of degree 3 or 4.