Quarter-turn flat three-manifold
= Quarter-turn flat three-manifold
The Euclidean transformations $a(x,y,z)=(x+1,y,z)$, $b(x,y,z)=(x,y+1,z)$ and $s(x,y,z)=(-y,x,z+1/4)$ generate a torsion-free cocompact group with $sas^{-1}=b$, $sbs^{-1}=a^{-1}$ and $s^4$ equal to unit translation along $z$. Every nontranslation has nonzero axial displacement modulo integers, hence no fixed point. Its maximal translation subgroup is $\mathbb Z^3$ and its holonomy is $C_4$.