In my study of Artin groups I need to compute the center of the group $G=\langle x,y\mid (xy)^l=(yx)^l\rangle$.
Of course, if $r= (xy)^l=(yx)^l$ then $yr=y(xy)^l=(yx)^ly=ry$ and $xr=x(yx)^l=(xy)^lx=rx$ and so $r\in Z(G)$ so it is obvious that $\langle r\rangle\subset Z(G)$ and from these one can deduce that $\langle xy\rangle\subset Z(G)$. However, I am not sure wheter this is an equality or I am missing something in the center.
Any help will be appreciated.
$$G/\langle r \rangle = \langle x,y \mid (xy)^l \rangle \cong \langle x,y \mid x^l \rangle,$$ which is the free product of an infinite cyclic group and a cyclic group of order $l$, and has trivial centre by the general theory of free products.
So $Z(G) = \langle r \rangle$.