Non-euclidean Affine Laminations

The purpose of the present paper is to discuss examples of aane Riemann surface laminations which do not admit a leafwise Euclidean structure. The rst example of such a lamination was constructed by Ghys Gh97]. Our discussion is based on the geometric methods developed by Lyubich, Minsky and the author LM97], KL01], which rely on the observation that any aane surface A gives rise in a natural way to a hyperbolic 3-manifold HA with a distinguished point at innnity. In particular, we give a new interpretation and a generalization of the example of Ghys. 1. Affine and hyperbolic laminations In this Section we recall the basic facts on the relationship between aane and hyperbolic laminations. Although our exposition is self-contained, more details on this relationship can be found in KL01]. 1.A. AAne and Euclidean surfaces. By endowing a Riemann surface S with an atlas of coordinate charts with transition maps from a given pseudo-group C (contained in the pseudo-group of all holomorphic maps) one can deene ner geometric structures on S. Deenition 1.1. We shall say that S is (i) an aane Riemann surface, if C is the group of all complex aane maps z 7 ! az + b; a; b 2 C ; a 6 = 0; (ii) a Euclidean surface, if C is the group of all maps z 7 ! az + b; a; b 2 C ; jaj = 1 (so that the transitions are Euclidean motions). If S is an aane surface, then its tangent and cotangent bundles (and hence all tensor bundles) are endowed with a natural at connection. Being parallel with respect to this connection means to have constant coeecients in any aane coordinate chart (the reader is referred to Ca88] and Go88] for general notions from the theory of aane manifolds). So, one can talk about parallel vector elds, forms, Riemannian metrics, etc. on S. In these terms a Euclidean surface is just an aane surface endowed with a parallel conformal metric. An aane Riemann surface structure is the same as a complex aane structure, or, in \real terms", a projective Euclidean (similarity) structure. In particular, an aane plane is R 2 endowed with the class of all multiples of a given Euclidean structure. Any complete aane surface is a quotient of the aane plane by a freely acting discrete group of Euclidean motions. Therefore, for any such surface the aane …