Free dihedral action on a drilled torus ($|D_n|=2n,g=2n+1$)

Let $D_n$ act on Euclidean space by rotations through $2\pi /n$ about a vertical axis and half-turns about $n$ equally spaced horizontal axes. A round solid torus about the vertical axis, drilled through radially along the $2n$ places where the horizontal axes cross its tube, has a smooth boundary of genus $2n+1$ after equivariant rounding of its rims. All nonidentity rotation axes lie in the complement of this boundary, so the induced group action is a free action of a group. Each bore replaces two closed discs by an annulus, lowering Euler characteristic by two and increasing genus by one. The boundary orientation is preserved; its induced Riemann surface has an unramified quotient space of genus two by the Riemann-Hurwitz formula.