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

Abstract

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.

Key Words:

• Block-diagonal matrices
• Free subgroup of rank two
• Infinite unitriangular matrices

2010 Mathematics Subject Classification:

• 20E05
• 20H25

Notes

Communicated by T. Lenagan.