By Danny Calegari
This particular reference, aimed toward study topologists, supplies an exposition of the 'pseudo-Anosov' idea of foliations of 3-manifolds. This concept generalizes Thurston's concept of floor automorphisms and divulges an intimate connection among dynamics, geometry and topology in three dimensions. major topics back to during the textual content comprise the significance of geometry, particularly the hyperbolic geometry of surfaces, the significance of monotonicity, specially in 1-dimensional and co-dimensional dynamics, and combinatorial approximation, utilizing finite combinatorical items comparable to train-tracks, branched surfaces and hierarchies to hold extra complex non-stop items.
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Extra info for Foliations and the Geometry of 3-Manifolds
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.