= Orbifold fundamental group
{title2=$\pi_1^{\rm orb}(B)$}
The <orbifold fundamental group> records loops together with their local isotropy relations, so a loop around a cone point of order $m$ has its $m$th power trivial rather than being trivial itself. For an oriented closed surface of <genus> $g$ with cone orders $m_1,\ldots,m_r$, its presentation is $\langle a_1,b_1,\ldots,a_g,b_g,c_1,\ldots,c_r\mid c_i^{m_i}=1,\ \prod_j[a_j,b_j]c_1\cdots c_r=1\rangle$. A torsion-free <finite-index subgroup> of a <hyperbolic surface> <orbifold> gives a smooth locally isometric surface cover. The absence of torsion means that the <subgroup> meets no nonidentity local cone stabilizer.
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