The Free Group F ₂

Abstract

In the previous chapter, we saw that every two Cohn matrices of a Cohn triple generate the same subgroup of SL($2, \mathbb{Z}$), namely the commutator subgroup SL($2, \mathbb{Z}$)’. This group turns out to have a very interesting structure leading to a new interpretation of the Cohn matrices as combinatorial words. On the way, we get to know the basics of free groups, prove a classical theorem about automorphisms of free groups, and see how the Cohn matrices can be used to solve an interesting problem describing primitive elements in the free group of rank 2.