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 [Invent. Math. 86 (1986) 287-346], 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 [Bull. Amer. Math. Soc. (N.S.) 6 (1982) 357-381].