Affine foliations and covering hyperbolic structures (Q5937281)

Thurston compactified the Teichmüller space of hyperbolic structures on a surface \(S\) by the space of projective measured laminations (or foliations) on \(S\). ``In this paper, we develop an analogous theory relating the space of affine laminations to another moduli space of geometric structures which contains Teichmüller space as a subset.'' ``Affine laminations are just being short of measured laminations: they are laminations whose lifts to the universal cover of the surface \(S\) are measured laminations, with the property that covering translations act on transversals by affine maps which stretch these transversals in a consistent way yielding stretch homomorphisms \(\Phi:\pi_1(S) \to \mathbb R_+\). Thus, in the surface \(S\), a transversal to the affine lamination carries a Borel measure which is only defined up to multiplication by a positive constant.'' ``We shall not only deal with affine laminations, but also with hyperbolic structures which are intimately related to affine laminations. These are hyperbolic structures on certain covers of \(S\) which also have good properties with respect to covering translations'' (fixing an ideal triangulation of a punctured surface \(S\), the covering translations are again stretch maps along the leaves of the lift of the ideal triangulation to the universal covering of \(S\)). ``The starting point of this work was the following question: Can the space of affine laminations serve as a boundary for some space of geometric structures on the surface? We answer the question completely for surfaces with punctures. The space of affine laminations is a fiber bundle over the cohomology space \(H^1(S, \partial S;\mathbb R)\) (\(\partial S\) being the set of punctures), whose elements are regarded as homomorphisms \(\log \Phi:\pi_1(S) \to \mathbb R\). Each fiber of the bundle is associated to a stretch homomorphism. Similarly, for a fixed ideal triangulation \(\lambda\), the space of covering hyperbolic structures forms a ``Teichmüller space'' which is a fiber bundle over the same base cohomology space. Each fiber of the Teichmüller space is compactified by a projectivized fiber of affine lamination space. When we restrict \(\Phi\) to be the trivial homomorphism, we recover Thurston's compactification of the Teichmüller space of hyperbolic structures on \(S\) by the space of projective measured foliations''.