By Robert C. Penner

There's an primarily “tinker-toy” version of a trivial package over the classical Teichmüller area of a punctured floor, referred to as the embellished Teichmüller house, the place the fiber over some extent is the gap of all tuples of horocycles, one approximately each one puncture. This version ends up in an extension of the classical mapping type teams known as the Ptolemy groupoids and to yes matrix versions fixing comparable enumerative difficulties, every one of which has proved beneficial either in arithmetic and in theoretical physics. those areas take pleasure in numerous comparable parametrizations resulting in a wealthy and complicated algebro-geometric constitution tied to the already complex combinatorial constitution of the tinker-toy version. certainly, the ordinary coordinates supply the prototypical examples not just of cluster algebras but additionally of tropicalization. This interaction of combinatorics and coordinates admits additional manifestations, for instance, in a Lie idea 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 offers the tale and wider context of those embellished Teichmüller areas as built through the writer during the last twenty years in a chain of papers, a few of them in collaboration. occasionally correcting mistakes or typos, occasionally simplifying proofs and occasionally articulating extra normal formulations than the unique learn papers, this quantity is self-contained and calls for little formal historical past. in accordance with a master’s path at Aarhus collage, it supplies the 1st therapy of those works in monographic shape.

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Extra resources for Decorated Teichmuller Theory

Example text

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. 1. There is a unique horocycle in H centered at each extended rational point x D Q [ f1g Â C, x and the horocycle centered at p 2 Q has Euclidean diameter Q q p 1 , where is written in reduced form except with n 2 Z written as n1 and 1 D ˙ 10 . 2 q q 24 1 The basics x are tangent to Furthermore, the horocycles in H centered at distinct points pq ; rs 2 Q one another if and only if ps qr D ˙1, and in this case, the horocycle in H tangent x to these two horocycles is centered at pCr 2 Q.

2. 2. h-length. Proof. 2 and the earlier calculation to arrange that the three u, be v, ab w. , x D p1 a . 2 be Now using 1 D h ; i, we find y D be uC p v D Dp 2be 2a a Â 1 , 2x so a be a be C ; 0; C 2be 2a 2be 2a Ã as a vector in R3 . Using the explicit mapping H ! x; y; z/ 7! z y x ; z 1 x /, we compute the imaginary part of the image of to be be ; since u, v, w respectively map a to 0; 1; 1, the hyperbolic segment lying in h. be v/ maps to the horizontal segment of a be a Euclidean length one at height a , which has hyperbolic length be using the expression for the hyperbolic metric in U.

0, y 7! 1, z 7! 1. x is real if and only if x, y, z, w lie on a common circle Furthermore, Œw; x; y; z 2 U x All these things can be routinely verified directly. or line in U. Given distinct ; Á 2 U, the hyperbolic distance between and Á is given by logŒ ; Á0 ; Á; 0 , where 0 , Á0 are the ideal endpoints of the geodesic connecting and Á enumerated so that lies between 0 and Á. Finally, exchanging any two of w, x, y, z leaves D Œw; x; y; z invariant, and the six possible remaining permutations of coordinates have cross ratios: ; 1 ; 1 ; 1 ; 1 1 ; 1 ; forming the group of anharmonic ratios.