Centre of orbifold fundamental group of torus (Klein bottle) with one cone point $\newcommand{\orb}{\mathrm{orb}}$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?
 A: 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.
A: The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a non-trivial center or not. If one wishes to be very explicit, one can even compute an explicit generating set for a finite index surface group inside the groups, see [2], particularly §5, where the Reidemeister-Schreier method is used to find a one-relator presentation on $4q-2$ generators for such a finite index subgroup.
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[1] Baumslag, G.; Taylor, Tekla, The centre of groups with one defining relator, Math. Ann. 175, 315-319 (1968). ZBL0157.34901.
[2] Baumslag, Gilbert; Troeger, Douglas, Virtually free-by-cyclic one-relator groups. I., Fine, Benjamin (ed.) et al., Aspects of infinite groups. A Festschrift in honor of Anthony Gaglione. Papers of the conference, Fairfield, USA, March 2007 in honour of Anthony Gaglione’s 60th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-279-340-9/hbk). Algebra and Discrete Mathematics (Hackensack) 1, 9-25 (2008). ZBL1188.20023.
A: 
The centre of any non-cyclic one-relator group with torsion
is trivial.

Moreover,

the centraliser of any non-identity element of a one-relator group with torsion
is cyclic.

This is B. B. Newman’s theorem (1973).
