Combinatorics of train tracks by R. C. Penner

By R. C. Penner

Measured geodesic laminations are a common generalization of straightforward closed curves in surfaces, they usually play a decisive position in quite a few advancements in two-and three-d topology, geometry, and dynamical structures. This booklet provides a self-contained and entire therapy of the wealthy combinatorial constitution of the gap of measured geodesic laminations in a hard and fast floor. households of measured geodesic laminations are defined by means of specifying a teach music within the floor, and the gap of measured geodesic laminations is analyzed through learning houses of teach tracks within the floor. the cloth is constructed from first ideas, the concepts hired are primarily combinatorial, and just a minimum historical past is needed at the a part of the reader. particularly, familiarity with straight forward differential topology and hyperbolic geometry is thought. the 1st bankruptcy treats the elemental thought of educate tracks as found via W. P. Thurston, together with recurrence, transverse recurrence, and the categorical development of a measured geodesic lamination from a measured educate tune. the following chapters enhance yes fabric from R. C. Penner's thesis, together with a traditional equivalence relation on measured teach tracks and conventional types for the equivalence periods (which are used to research the topology and geometry of the gap of measured geodesic laminations), a duality among transverse and tangential buildings on a educate tune, and the specific computation of the motion of the mapping classification workforce at the area of measured geodesic laminations within the floor.

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K72537. ) 46 K. Bezdek Blaschke–Lebesgue problems. In this way, ball-polyhedra are investigated as well. The rest of the paper studies the topics outlined in six consecutive sections. 2. Plank Theorems A convex body of the Euclidean space Ed is a compact convex set with non-empty interior. Let C ⊂ Ed be a convex body, and let H ⊂ Ed be a hyperplane. Then the distance w(C, H) between the two supporting hyperplanes of C parallel to H is called the width of C parallel to H. Moreover, the smallest width of C is called the minimal width of C and is denoted by w(C).

Bezdek and R. Schneider, Covering large balls with convex sets in spherical space, Beitrage ¨ zur Alg. und Geom. 51/1 (2010), 229–235. [18] W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Math. Ann. 76 (1915), 504–513. [19] T. Bonnesen and W. Fenchel, Theory of Convex Bodies, (English translation), BCS Associates (Moscow, Idaho, USA), 1987. [20] P. Brass, W. Moser and J. Pach, Research problems in discrete geometry, Springer, New York, 2005. [21] S. Campi, A. Colesanti and P.

Then n r(Ci ∩ B) ≥ r(B). 1 can be found in [12]. 1 in Banach spaces. Thus, an affirmative answer to the following problem would improve the plank theorem of Ball. 2. Let the ball B be covered by the convex bodies C1 , C2 , . . , Cn in an arbitrary Banach space. Prove or disprove that n r(Ci ∩ B) ≥ r(B). i=1 It is well-known that Bang’s plank theorem holds in complex Hilbert spaces as well. However, for those spaces Ball [4] was able to prove the following much stronger theorem. 3. If the planks of widths w1 , w2 , .

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