Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that $π_1(\operatorname{Cay}(G, S)) = ⟨grg^{−1}⟩_{g∈G,r∈R}$.
They are here. The paragraph above 8.1 on page 49.
http://www.math.ethz.ch/~alsisto/LectureNotesGGT.pdf
My problem is I looked the example $\mathbb{Z}\times \mathbb{Z} = \langle a,b\mid aba^{-1}b^{-1}\rangle$. I do not see how I can find an element in $\langle grg^{−1}\rangle_{g∈G,r∈R}$ that equals say $a^{-2}ba^2b^{-1}$.
Thanks for any help.