By Robert C. Penner

There's an primarily “tinker-toy” version of a trivial package deal over the classical Teichmüller house of a punctured floor, known as the adorned Teichmüller house, the place the fiber over some degree is the gap of all tuples of horocycles, one approximately every one puncture. This version results in an extension of the classical mapping classification teams known as the Ptolemy groupoids and to yes matrix types fixing comparable enumerative difficulties, every one of which has proved invaluable either in arithmetic and in theoretical physics. those areas take pleasure in a number of comparable parametrizations resulting in a wealthy and complex algebro-geometric constitution tied to the already complex combinatorial constitution of the tinker-toy version. certainly, the normal coordinates provide 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 thought 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 means of the writer over the past twenty years in a sequence of papers, a few of them in collaboration. occasionally correcting blunders or typos, occasionally simplifying proofs and infrequently articulating extra basic formulations than the unique learn papers, this quantity is self-contained and calls for little formal heritage. in line with a master’s direction at Aarhus collage, it supplies the 1st therapy of those works in monographic shape.

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There's an basically “tinker-toy” version of a trivial package over the classical Teichmüller area of a punctured floor, referred to as the embellished Teichmüller area, the place the fiber over some extent is the distance of all tuples of horocycles, one approximately every one puncture. This version ends up in an extension of the classical mapping classification teams known as the Ptolemy groupoids and to yes matrix types fixing comparable enumerative difficulties, each one of which has proved valuable either in arithmetic and in theoretical physics.

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**Example text**

1. 1. Lambda length. 33 4 Basic definitions and formulas These are our basic invariants, and essentially all of our calculations will be performed using them. ) Put another way, a decoration on a hyperbolic geodesic is the specification of a pair of horocycles, one centered at each of its ideal points, and the lambda length is an invariant of a decorated geodesic. Thus though a hyperbolic geodesic has infinite length, a decoration gives a way to truncate it and thereby define a sensible finite length ı, essentially the lambda length.

Regions bounded by three disjoint geodesics pairwise sharing ideal points at infinity. 3. 3, where is regarded as a set of geodesics. Z/ consisting of all two-by-two integral matrices of determinant one again modulo the equivalence relation identifying the matrix A with its negative A. Z/) is called the (classical) modular group and plays a basic role in number theory, as we shall partly explain in this section. It is also among the simplest of mapping class groups, namely, the mapping class group of F11 (where we have taken the quotient by the hyperelliptic involution).

0; v 7! 1; w 7! , the determinant is positive. The further cases where one of u, v, w might be 1 are handled similarly. It follows that the fractional linear transformation Au01;v0 ;w 0 Au;v;w maps u 7! u0 , v 7! v 0 and w 7! R/ indeed acts transitively. , we have b D 0; d aCb D 1; cCd and c D 0; from which it follows that B D ˙I . R/: • if ja C d j < 2, then A is said to be elliptic, and there is a unique fixed point in U with A described as rotation about this fixed point; • if ja C d j D 2, then A is said to be parabolic, and there is a unique fixed point at infinity; • if ja C d j > 2, then A is said to be hyperbolic, and there is a pair of fixed points at infinity, which are the ideal points of a unique geodesic in U with A acting as translation along this geodesic.