By Peter Field

An Unabridged, Digitally Enlarged Printing, to incorporate All Figures; A direction In Projective Geometry Which Emphasizes The Technical purposes: Definitions And primary Notions - airplane Homology - Linkages - the entire Quadrangle And Quadrilateral - The Cross-Ratio - levels And Pencils - The Hexagon - Involution - Pole And Polar - The Null method - accomplished Index

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Remark We used a hyperbolic structure on to construct S1∞ ( ) together with the action of π1 ( ) on it. In fact, it is possible to construct this space directly from a topological surface S. Given an essential simple closed curve α ⊂ S, form the preimage α in S ≈ R2 . The set of ends E of components of α admits a natural circular ordering which comes from the embedding of α in the plane, and which is preserved by the deck group of S. A circularly ordered set admits a natural topology, called the order topology, which we will deﬁne and study in Chapter 2.

Any ﬁnite string which appears in S∞ appears with density bounded below by some positive constant. 2. The proportion of 2’s in Sn is at least r for n odd and at most 1 − r for n even. Let be a genus 2 surface, obtained as the union of two 1-holed tori T1 , T2 . Let r be an inﬁnite geodesic ray in obtained from S∞ as a union of loops in the Ti representing (1, 1) curves, where the ﬁrst two loops are in T2 , then one loop in T1 , and so on according to the “code” S∞ . Then r can be pulled tight to a unique geodesic ray, with respect to any hyperbolic structure on .

Gluing polygons) Let be a hyperbolic surface of genus g. Pick a point p ∈ and a conﬁguration of 2g geodesic arcs with endpoints at p which cut up into a hyperbolic 4g-gon P. e. the angles are determined by the lengths. An n-gon with edge lengths assigned has n − 3 degrees of freedom. The polygon P satisﬁes extra constraints: the edges are glued in pairs, so there are only 2g degrees of freedom for the edges. Moreover, the sum of the angles is 2π , so there are 4g − 4 degrees of freedom for the angles.