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In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R 3 . M of finite topology means M is homeomorphic to a compact surface M (of genus k and empty boundary) minus a finite number of points p 1 , ..., p j ∈ M , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends sufficiently small so they are topologically S 1 × [0, 1) and hence, annular. We remark that M is orientable since M is properly embedded in R 3 .The simplest examples (discovered by Meusnier in 1776) are the helicoid and catenoid (and a plane of course). It was only in 1982 that another example was discovered. In his thesis at Impa, Celso Costa wrote down the Weierstrass representation of a complete minimal surface modelled on a 3-punctured torus. He observed the three ends of this surface were embedded: one top catenoidtype end 1 , one bottom catenoid-type end, and a middle planar-type end 2 [8]. Subsequently, Hoffman and Meeks [15] proved this example is embedded and they constructed for every finite positive genus k embedded examples of genus k and three ends.In 1993, Hoffman, Karcher and Wei [14] discovered the Weierstrass data of a complete minimal surface of genus one and one annular end. Computer generated pictures suggested this surface is embedded and the end is asymptotic to an end of a helicoid. Hoffman, Weber and Wolf [17] have now given a proof that there is such an embedded surface. Moreover, computer evidence suggests that one can add an arbitrary finite number k of handles to a helicoid to obtain a properly embedded genus k minimal surface asymptotic to a helicoid.For many years, the search went on for simply connected examples other than the plane and helicoid. We shall prove that there are no such examples.

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 *

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