No abstract

No abstract

No abstract

No abstract

If G is a finite group acting smoothly on a closed surface F, it is well known that G leaves invariant some Riemannian metric of constant curvature on F. Thus any action of G on the 2-sphere S 2 is conjugate in Diff(S 2) to an orthogonal action. If G acts on the torus SX• S 1, there is a G-invariant flat metric on S a • S 1, and if G acts on a surface F with negative Euler number, then F admits a G-invariant hyperbolic metric.Recently Thurston, [Th 1, Th 2, Th 3], has described the eight 3-dimensional geometries which provide geometric structures for closed 3-manifolds in the same way that the 2-sphere S 2, the Euclidean plane E 2 and the hyperbolic plane H 2 provide geometric structures for surfaces. See also the survey article by Scott [Sc4]. Thurston also conjectured that if M is a closed 3-manifold with a geometric structure modelled on one of these eight geometries, say X, then any smooth action of a finite group G on M should leave invariant some metric on M inducing the geometry X. We will say that G preserves the geometric structure on M in this case. It should be noted that the restriction to smooth actions of G on M is essential. For Bing [Bi] showed that there are involutions of S 3 whose fixed set is a wild 2-sphere. However, in dimension two, it was proved by Eilenberg [Ei] that any action of a finite group on a surface is conjugate to a smooth action.In this paper, our main result asserts that Thurston's conjecture holds for five of the eight geometries. The result is the following.Theorem2.1. Let M be.a closed 3-manifold with a geometric structure modelled on one of H 2 • ~., SL 2 n~., Nil, E 3 or Sol. Then any smooth finite group action on M preserves the geometric structure on M.Our methods yield no information about group actions on manifolds modelled on any of the three remaining geometries S 3, SEx R and H 3. However, Meeks and Yau [M-Y3] have verified Thurston's conjecture for manifolds modelled on SEx ~ except in the case when G contains the alternating group A 5 9 Thurston [Th4] has announced some results which overlap substantially with the results of this paper. He shows that if M is a closed 3-manifold with *