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In this paper, we introduce a method for dividing up a noncompact hyperbolic manifold of finite volume into canonical Euclidean pieces. The construction first arose in the setting of surfaces (see [7]), and in this case one gets a canonical cell decomposition of the surface and a canonical Euclidean structure. (The Euclidean structure, of course, is not complete.) The conformal structure underlying this Euclidean structure does not agree with the underlying hyperbolic structure, but the two conformal structures are probably not too distant (cf. Sullivan's theorem [5] for an analogous result).This investigation arose from an attempt to understand the coordinates and cell decomposition of Teichmϋller space due to Harer and Mumford [6] and independently to Thurston. Such coordinates and cell decompositions are also provided in [3] and [7]; in the latter, the action of the mapping class group on the coordinates is considered. We would like to thank J. Harer for the inspiration of his work and for several helpful remarks.Our method is to work in Minkowski space and to represent a cusp by a point on the light-cone. The orbit of this point turns out to be discrete (even though the action of the group on the light-cone is ergodic), and we take the convex hull of the orbit. The boundary of this convex hull is decomposed into affine pieces, and one should think of the convex hull boundary as a kind of piecewise linear approximation to the upper sheet of the hyperboloid in Minkowski space. Each piece has a natural Euclidean structure. The suggestion that this might be possible first arose in a conversation between the authors and Lee Mosher. We thank Mosher for his contribution to this crucial idea. Comments by Brian Bowditch have also been helpful on a number of occasions. As a final credit, we wish to thank Bill Thurston. Much of this work as been discussed at various points with him, and the exposition has gained substantially from his comments.

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The photosynthetic light reactions of green plants are mediated by chlorophyll-binding protein complexes located in the thylakoid membranes within the chloroplasts. Thylakoid membranes have a complex structure, with lateral segregation of protein complexes into distinct membrane regions known as the grana and the stroma lamellae. It has long been clear that some protein complexes can diffuse between the grana and the stroma lamellae, and that this movement is important for processes including membrane biogenesis, regulation of light harvesting, and turnover and repair of the photosynthetic complexes. In the grana membranes, diffusion may be problematic because the protein complexes are very densely packed (approximately 75% area occupation) and semicrystalline protein arrays are often observed. To date, direct measurements of protein diffusion in green plant thylakoids have been lacking. We have developed a form of fluorescence recovery after photobleaching that allows direct measurement of the diffusion of chlorophyll-protein complexes in isolated grana membranes from Spinacia oleracea. We show that about 75% of fluorophores are immobile within our measuring period of a few minutes. We suggest that this immobility is due to a protein network covering a whole grana disc. However, the remaining fraction is surprisingly mobile (diffusion coefficient 4.6 6 0.4 3 10 211 cm 2 s 21 ), which suggests that it is associated with mobile proteins that exchange between the grana and stroma lamellae within a few seconds. Manipulation of the protein-lipid ratio and the ionic strength of the buffer reveals the roles of macromolecular crowding and protein-protein interactions in restricting the mobility of grana proteins.