I have calculated the fundamental group of the annulus and got the following group presentation:
$ \langle a, b | ab = ba = 1 \rangle$
This is the set of strings of the form: $1, a, a^2, a^3, \dots , b , b^2 , \dots$.
Is this equivalent to $\langle a | \rangle = \mathbb{Z}$? If yes, how do I see that?
Edit I think it's not equivalent. : (
Many thanks for your help!