Isometric free finite group actions on non-positively curved 3-manifolds

Let $M$ be a closed orientable $3$-manifold admitting a metric of nonpositive sectional curvature (an NPC metric), and let $G$ be a finite group acting freely on $M$ by orientation-preserving diffeomorphisms.

Previous results showed that $M$ admits a $G$-invariant NPC metric except possibly when $M$ is a graph manifold.

In this paper, we resolve the remaining case by proving that $M$ also admits a $G$-invariant NPC metric when $M$ is a graph manifold.

This result advances our understanding in dimension $3$ of the question posed by Schoen-Yau about Nielsen realization for NPC $3$-manifolds.