By Robert C. Penner
There's an basically “tinker-toy” version of a trivial package deal over the classical Teichmüller house of a punctured floor, referred to as the embellished Teichmüller area, the place the fiber over some degree is the distance of all tuples of horocycles, one approximately every one puncture. This version ends up in an extension of the classical mapping category teams referred to as the Ptolemy groupoids and to convinced matrix versions fixing similar enumerative difficulties, every one of which has proved beneficial either in arithmetic and in theoretical physics. those areas take pleasure in a number of comparable parametrizations resulting in a wealthy and complicated algebro-geometric constitution tied to the already tricky combinatorial constitution of the tinker-toy version. certainly, the normal coordinates supply 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 conception 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 provides the tale and wider context of those adorned Teichmüller areas as constructed by way of the writer over the past twenty years in a chain of papers, a few of them in collaboration. occasionally correcting error or typos, occasionally simplifying proofs and occasionally articulating extra common formulations than the unique study papers, this quantity is self-contained and calls for little formal historical past. in response to a master’s direction at Aarhus collage, it provides the 1st therapy of those works in monographic shape.
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4 and the definition of continued fractions) is left to the reader. 4, we may identify these two sets. Z/, then we let A act on the right on the points at infinity (following Gauss’ convention in fact), and hence on oriented geodesics, and shall thus identify an oriented edge of with the image under the right action of A of the distinguished oriented edge from 0 to 1. 3. Furthermore, reversing the orientation on an edge labeled A corresponds to re-labeling SA. Another number-theoretic aspect of the current discussion involves so-called elliptic curves, namely, discrete subgroups ƒ of C of rank 2, also called lattices.
Minkowski three-space. x; y; z/ 2 R3 W hu; ui D C1g: Notice that if u; v 2 LC , then hu; vi Ä 0 with equality if and only if u is a scalar multiple of v, and if u 2 LC ; v 2 H, then hu; vi < 0, where both inequalities follow from the Cauchy–Schwarz inequality. LC [ LC / has the corresponding attribute. One checks without difficulty that a plane … D fu 2 R3 W hu; vi D g with Minkowski normal v, for some 2 R, is elliptic, parabolic, or hyperbolic, respectively, if and only if hv; vi is negative, vanishes, or is positive.
Decorated Teichmuller Theory by Robert C. Penner
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