In Meeks and Scott’s paper, Finite Group Actions on 3-Manifolds, they assert that if a finite group $ G $ acts on $ F \times \mathbb{S}^{1} $ (where $ F $ is a surface with boundary) while preserving the Seifert fibration up to homotopy, then the product structure on the boundary of $ F \times \mathbb{S}^{1} $ can be homotoped to be $ G $-invariant. They claim that this follows from the fact that $ G $ preserves a flat metric on the boundary.
I understand fully why it preserves a flat metric on the boundary (from lifting the flat metric on the quotient boundary), but I don’t see exactly how to show that this implies their claim.
The reference for this claim is below Theorem 2.3 on Page 299.
Well, I think if the group action preserve a flat metric on the boundary, the action is by isometries, so it maps geodesics to geodesics. The Seifert fibers on the boundary determine a single homotopy class (think of it as a topological direction) and there are simple closed geodesics in that homotopy class. In fact there is a whole foliation of the boundary surface into simple closed geodesics, each freely homotopic to any of the original Seifert fibers. This foliation is invariant with respect to the group action. Now you isotope the Seifert fibration of the manifold in a small "tubular" neighborhood of the boundary surface, while outside that neighborhood it is identity. And you do it so that the original fibers on the boundary are isotoped to be the geodesic foliation on the soudary surface.
Does that make sense?