= Freeness criterion for a subgroup of a polygon-cover deck group
{title2=$U\cap g\langle C_j\rangle g^{-1}=\{1\}$}
In the <paired-polygon construction of finite surface covers>, a <subgroup> $U$ acts freely precisely when the displayed intersection is trivial for every vertex word $C_j$ and every $g\in T$. The quotient map $M\to U\backslash M$ is then a <normal covering map> with deck <group> $U$ and $\chi(U\backslash M)=\chi(M)/|U|$. Normality of $U$ in $T$ is instead the criterion for the intermediate <orbifold> cover $U\backslash M\to T\backslash M$ to be normal. These are different <covering maps>.
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