$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,


$$\pi\_1^{\orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$


$$\pi\_1^{\orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$


Want to know the centre of these two groups. Are they trivial?



Centre of orbifold fundamental group of torus (Klein bottle) with one cone point
These groups have trivial centers. As one proof, they are both fuchsian and so embed in $\mathrm{PSL}(2, \mathbb{R})$ (well, the orientation preserving subgroups do). However, elements of $\mathrm{PSL}$ that commute must have common fixed points on the "circle at infinity" of the hyperbolic plane.


