Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3–manifolds

Abstract

We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3‐manifolds. Our main result is that such actions on elliptic and hyperbolic 3‐manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3‐manifolds (in the sense of Thurston) are always geometric, ie there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston [32]. 57M60, 57M50; 53C21, 53C44