Those groups r which act properly discontinuously and aillnely on II?' with compact fundamental domain are classified. First it is shown that such a group f contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive alTine actions on IR' are classified; this leads to a classification of atTine crystallographic groups acting on IR3. A characterization of which abstract groups admit such an action is given; moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups (respectively simply transitive afline groups) is induced by conjugation by a polynomial automorphism of the affrne space. A characterization is given of which closed 3-manifolds can be represented as quotients of IR' by groups of afftne transformations: a closed 3-manifold M admits a complete atline structure if and only if M has a finite covering homeomorphic (or homotopy-equivaient) to a 2-torus bundle over the circle.

For X a closed, oriented hyperbolic manifold and p an orthogonal representation of ~IX, we define a Ruelle-type zeta function Rp(s) for Re(s) large as the product over prime closed geodesics 7 of factors det(l-p(7)e-~l;)), where l (7 ) is the length of 7. We analytically continue Rp(s) and compute its leading term at s=0. In odd dimensions this gives a formula for the Ray-Singer analytic torsion Tp(X) in terms of closed geodesics. This shows that the periodic orbits of a hyperbolic geodesic flow can be used to compute Reidemeister torsion via a zeta function in analogy to our earlier results for other classes of hyperbolic flows.In an appendix we compute the functional determinant of the Laplacian on a hyperbolic surface.Let K be a finite complex and p a representation of n t K by real matrices of determinant _+1. Reidemeister torsion also has a dynamical interpretation in many cases IF2]. For a flow on a closed manifold with circular chain recurrent set we showed in [FI] that z can be computed from its closed orbits using a suitable zeta function. This inchtdes flows with cross-section and Smale flows. The case of Smale flows on S 3 was first treated (with different terminology) by Franks, who showed how to compute the Alexander polynomial of a link from a suitable flow on its exterior [Fr]. These results depend on applying the Lefschetz fixed point formula to suitably constructed transversals to the flow. Here y runs over the prime closed geodesics of X (i.e. those not expressible as multiples oJ some shorter closed geodesic) and l(7 ) denotes the le'ngth oJ 7.In [F2] we compare this to our earlier results and give analogous theorems for geodesic flows on some other Riemannian manifolds. In particular, the exponent e, is explained as being the Poincar6 index of the periodic orbits of the geodesic flow on a hyperbolic d-manifold: the right zeta function for computing torsion is R ~.The proof of this theorem begins by expressing Rp in terms of certain Selberg-type zeta functions S o ..... Sd-1. Here each S i is characterized by the property that its logarithmic derivative Y~=S'I/S i is a certain sum over closed geodesics that arises when the Selberg Trace Formula is applied to i-dimensional twisted forms. One has Si=Sd_l_ i and each Si satisfies a certain functional equation relating its values at s and d-l-s.These properties alone allow one to prove Theorem 1 for d=2 IF5] but for d>2 more analysis is needed.We recall that Ray and Singer defined an analytic torsion To(M ) for every closed Riemannian manifold X and orthogonal representation p of ~zlX [RSI]. Their definition involves the spectrum of the Laplacian on twisted forms. They conjectured, and it was proved by Cheeger I-C] and Miiller [Mu], that when p is acyclic and orthogonal Tp(X)= %(X). Our proof of Theorem 1 for d>2 will use this result. We have 7zI(SX) ~IX for d>2, so p in Theorem 1 may be identified with a representation of 7zlX. Then topology gives zp(SX)=zv(X) 2 (see Section 1). We then need to relate Tp(X) 2 to Rp(0). But Rp is expressible by ...

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