## Geometry & Topology

### 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].

#### Article information

Source
Geom. Topol., Volume 13, Number 2 (2009), 1129-1173.

Dates
Revised: 9 January 2009
Accepted: 28 November 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800226

Digital Object Identifier
doi:10.2140/gt.2009.13.1129

Mathematical Reviews number (MathSciNet)
MR2491658

Zentralblatt MATH identifier
1181.57023

#### Citation

Dinkelbach, Jonathan; Leeb, Bernhard. Equivariant Ricci flow with surgery and applications to finite group actions on geometric $3$–manifolds. Geom. Topol. 13 (2009), no. 2, 1129--1173. doi:10.2140/gt.2009.13.1129. https://projecteuclid.org/euclid.gt/1513800226

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