By Robert C. Penner
There's an primarily “tinker-toy” version of a trivial package over the classical Teichmüller area of a punctured floor, known as the embellished Teichmüller area, the place the fiber over some degree is the gap of all tuples of horocycles, one approximately every one puncture. This version ends up in an extension of the classical mapping type teams referred to as the Ptolemy groupoids and to sure matrix versions fixing comparable enumerative difficulties, every one of which has proved worthwhile either in arithmetic and in theoretical physics. those areas get pleasure from numerous similar parametrizations resulting in a wealthy and complex algebro-geometric constitution tied to the already complicated combinatorial constitution of the tinker-toy version. certainly, the typical coordinates provide the prototypical examples not just of cluster algebras but in addition of tropicalization. This interaction of combinatorics and coordinates admits extra manifestations, for instance, in a Lie concept for homeomorphisms of the circle, within the geometry underlying the Gauss product, in profinite and pronilpotent geometry, within the combinatorics underlying conformal and topological quantum box theories, and within the geometry and combinatorics of macromolecules.
This quantity supplies the tale and wider context of those embellished Teichmüller areas as built by way of the writer during the last 20 years in a chain of papers, a few of them in collaboration. occasionally correcting error or typos, occasionally simplifying proofs and occasionally articulating extra basic formulations than the unique learn papers, this quantity is self-contained and calls for little formal heritage. in keeping with a master’s direction at Aarhus college, it offers the 1st therapy of those works in monographic shape.
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Additional resources for Decorated Teichmuller Theory
R/. an indefinite form Œa; b; c is the hyperbolic geodesic connecting the two real roots of az 2 C bz C c D 0. 4 Basic definitions and formulas This section is principally based on  but also includes calculations from , , . Throughout, we shall conveniently normalize using the following result. 1. Given three distinct rays r1 ; r2 ; r3 Â LC from the origin, there are unique uj 2 rj so that huj ; uk i D 1 for j ¤ k, j; k D 1; 2; 3. Proof. Choose any vj 2 rj for j D 1; 2; 3. 2.
There is a unique horocycle contained in such a triangular region which is tangent to hn and hnC1 as well as tangent to the real axis, and we let hnC 1 denote this horocycle, which is evidently tangent to the 2 real axis at the half-integer point n C 12 and of Euclidean diameter 14 . We may continue recursively in this manner, adding new horocycles tangent to the real axis and tangent to pairs of consecutive tangent horocycles in order to produce a family of horocycles H in U. 1. 1. Horocyclic packing of U.
Then the signed volume of the tetrahedron determined by these four points is given by Ä 2 2 2 2 p C 223 C 214 13 13 ; C 34 2 2 12 23 34 14 12 12 23 13 34 14 13 where the volume is positive if and only if the Euclidean segment connecting u1 ; u3 lies below the Euclidean segment connecting u2 , u4 . Proof. 13.
Decorated Teichmuller Theory by Robert C. Penner