Affine lamination spaces for surfaces (Q1207879)

A lamination \(L\) in a compact surface \(M\) (possibly with a boundary) is said to be affine if it admits a transverse affine structure, i.e. a projective class of transverse measures on the lift \(\widetilde {L}\) in the universal cover \(\widetilde {M}\) for which the deck transformations act as scalar multiplication. \({\mathcal {AL}}(M)\) is the set of isotopy classes of non-empty affine laminations in \(M\) without Reeb and \(\partial\)-Reeb components. The main result is that \({\mathcal {AL}}(M)\), equipped with a natural topology, is homeomorphic to the product \({\mathcal {PL}}(M)\times H^ 1(M,\mathbb{R})\), \({\mathcal {PL}}(M)\) being the space of projective classes of measured laminations considered by \textit{W. Thurston} [Bull. Am. Math. Soc., New Ser. 19, No. 2, 417-431 (1988; Zbl 0674.57008)].