On Poincaré’s Theorem for Fundamental Polygons
نویسنده
چکیده
Poincaré’s classical theorem of fundamental polygons is a widely known, valuable tool that gives sufficient conditions for a (convex) hyperbolic polygon, equipped with so-called side-pairing transformations, to be a fundamental domain for a discrete subgroup of isometries. Poincaré first published the theorem in dimension two in 1882. In the past century, there have been several published proofs of this theorem (though many of them are questionably valid). It is the goal of this paper to present a proof of Poincaré’s Fundamental Polygon Theorem. We first examine the relevant hyperbolic geometry.
منابع مشابه
SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS
In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for ...
متن کاملPoincaré’s Theorem for Fuchsian Groups
We present a proof of Poincaré’s Theorem on the existence of a Fuchsian group for any signature (g; m1, ..., mr), where g, m1, ..., mr provide an admissable solution to the formula describing the hyperbolic area of the group’s quotient space. Along the way we elucidate relevant concepts in hyperbolic geometry and the theory of Fuchsian groups.
متن کاملFundamental Polygons by Maurice Heins
A group © satisfying the hypothesis of Siegel is necessarily of the first kind. The question arises whether there is a theorem of the Siegel type for © of the second kind. I t is one of the objects of the present investigation to establish such a theorem and to draw consequences of this result taken in conjunction with Siegel's theorem. The other object is to study the relation between the para...
متن کاملVector and scalar potentials, Poincaré’s theorem and Korn’s inequality
In this Note, we present several results concerning vector potentials and scalar potentials in a bounded, not necessarily simply-connected, three-dimensional domain. In particular, we consider singular potentials corresponding to data in negative order Sobolev spaces. We also give some applications to Poincaré’s theorem and to Korn’s inequality.
متن کاملDouble-null operators and the investigation of Birkhoff's theorem on discrete lp spaces
Doubly stochastic matrices play a fundamental role in the theory of majorization. Birkhoff's theorem explains the relation between $ntimes n$ doubly stochastic matrices and permutations. In this paper, we first introduce double-null operators and we will find some important properties of them. Then with the help of double-null operators, we investigate Birkhoff's theorem for descreate $l^p$ sp...
متن کامل