Geometry & Topology 13 (2009) 1129–1173

DOI: 10.2140/gt.2009.13.1129

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].

Keywords

group action, Ricci flow, geometric manifold

Mathematical Subject Classification

Primary: 57M50, 57M60

Secondary: 53C21, 53C44

References

Publication

Received: 7 July 2008
Revised: 9 January 2009
Accepted: 28 November 2008
Published: 27 January 2009
Proposed: Dave Gabai
Seconded: Colin Rourke, Tobias Colding

Authors