Conjugacy Classes in $\mathbb{Z}*\mathbb{Z}$?

Question

May I know what are the conjugacy classes of the group $\mathbb{Z}*\mathbb{Z}$ (the free product of infinite cyclic groups by itself)?

Answer 1

This group is more commonly known as the free group of rank $2$.

Every element of a free group is conjugate to a cyclically reduced word, and two cyclically reduced words are conjugate if and only if one is a cyclic permutation of the other. So it is easy to decide when two elements are conjugate.

For example, in the free group with generators $a$ and $b$, the element $aba^2b^{-1}ab^{-1}a^{-1}$ is conjugate to the cyclically reduced word $a^2b^{-1}a$, which is conjugate to $ab^{-1}a^2$, $b^{-1}a^3$ and $a^3b^{-1}$.