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Concept# Fundamental domain

Summary

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.
Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain for this action is a set D of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in X with a set of measure zero, for a certain (quasi)invariant measure on X. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the orbits. Frequently D is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation in ergodic theory. If a fundamental domain is used to calculate an integral on X/G, sets of measure zero do not matter.
For example, when X is Euclidean space Rn of dimension n, and G is the lattice Zn acting on it by translations, the quotient X/G is the n-dimensional torus. A fundamental domain D here can be taken to be [0,1)n, which differs from the open set (0,1)n by a set of measure zero, or the closed unit cube [0,1]n, whose boundary consists of the points whose orbit has more than one representative in D.
Examples in the three-dimensional Euclidean space R3.

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In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. A fundamental pair of periods is a pair of complex numbers such that their ratio is not real. If considered as vectors in , the two are not collinear. The lattice generated by and is This lattice is also sometimes denoted as to make clear that it depends on and It is also sometimes denoted by or or simply by The two generators and are called the lattice basis.

Fundamental domain

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral.

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Let G be a connected reductive algebraic group over an algebraically closed field k,gamma is an element of g( k(( epsilon ))) a semisimple regular element, we introduce a fundamental domain F gamma for the affine Springer fibers X gamma. We show that the purity conjecture of X gamma is equivalent to that of F gamma via the Arthur-Kottwitz reduction. We then concentrate on the unramified affine Springer fibers for the group GL(d). It turns out that their fundamental domains behave nicely with respect to the root valuation of gamma. We formulate a rationality conjecture about a generating series of their Poincare polynomials, and study them in detail for the group GL(3). In particular, we pave them in affine spaces and we prove the rationality conjecture.

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

The goal of this document is to provide a generalmethod for the computational approach to the topology and geometry of compact Riemann surfaces. The approach is inspired by the paradigms of object oriented programming. Our methods allow us in particular to model, for numerical and computational purposes, a compact Riemann surface given by Fenchel-Nielsen parameters with respect to an arbitrary underlying graph, this in a uniformand robust manner. With this programming model established we proceed by proposing an algorithmthat produces explicit compact fundamental domains of compact Riemann surfaces as well as generators of the corresponding Fuchsian groups. In particular, we shall explain how onemay obtain convex geodesic canonical fundamental polygons. In a second part we explain in what manner simple closed geodesics are represented in our model. This will lead us to an algorithm that enumerates all these geodesics up to a given prescribed length. Finally, we shall briefly overview a number of possible applications of our method, such as finding the systoles of a Riemann surface, or drawing its Birman-Series set in a fundamental domain.