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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kleinian group ： ウィキペディア英語版 Kleinian group In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C). The group PSL(2, C) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H3, and as orientation preserving conformal maps of the open unit ball ''B''3 in R3 to itself. Therefore a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces. There are some variations of the definition of a Kleinian group: sometimes Kleinian groups are allowed to be subgroups of PSL(2, C).2 (PSL(2, C) extended by complex conjugations), in other words to have orientation reversing elements, and sometimes they are assumed to be finitely generated, and sometimes they are required to act properly discontinuously on a non-empty open subset of the Riemann sphere. A Kleinian group is said to be of type 1 if the limit set is the whole Riemann sphere, and of type 2 otherwise. The theory of general Kleinian groups was founded by and , who named them after Felix Klein. The special case of Schottky groups had been studied a few years earlier, in 1877, by Schottky. ==Definitions== By considering the ball's boundary, a Kleinian group can also be defined as a subgroup Γ of PGL(2,C), the complex projective linear group, which acts by Möbius transformations on the Riemann sphere. Classically, a Kleinian group was required to act properly discontinuously on a non-empty open subset of the Riemann sphere, but modern usage allows any discrete subgroup. When Γ is isomorphic to the fundamental group $\backslash pi\_1$ of a hyperbolic 3-manifold, then the quotient space H3/Γ becomes a Kleinian model of the manifold. Many authors use the terms ''Kleinian model'' and ''Kleinian group'' interchangeably, letting the one stand for the other. Discreteness implies points in ''B''3 have finite stabilizers, and discrete orbits under the group ''G''. But the orbit ''Gp'' of a point ''p'' will typically accumulate on the boundary of the closed ball $\backslash bar^3$. The boundary of the closed ball is called the ''sphere at infinity'', and is denoted $S^2\_\backslash infty$. The set of accumulation points of ''Gp'' in $S^2\_\backslash infty$ is called the ''limit set'' of ''G'', and usually denoted $\backslash Lambda(G)$. The complement $\backslash Omega(G)=S^2\_\backslash infty\; -\; \backslash Lambda(G)$ is called the domain of discontinuity or the ordinary set or the regular set. Ahlfors' finiteness theorem implies that if the group is finitely generated then $\backslash Omega(G)/G$ is a Riemann surface orbifold of finite type. The unit ball ''B''3 with its conformal structure is the Poincaré model of hyperbolic 3-space. When we think of it metrically, with metric :$ds^2=\; \backslash frac$ it is a model of 3-dimensional hyperbolic space H3. The set of conformal self-maps of ''B''3 becomes the set of isometries (i.e. distance-preserving maps) of H3 under this identification. Such maps restrict to conformal self-maps of $S^2\_\backslash infty$, which are Möbius transformations. There are isomorphisms :$\backslash mbox(S^2\_\backslash infty)\; \backslash cong\; \backslash mbox(B^3)\; \backslash cong\; \backslash mbox(\backslash mathbf^3).$ The subgroups of these groups consisting of orientation-preserving transformations are all isomorphic to the projective matrix group: PSL(2,C) via the usual identification of the unit sphere with the complex projective line P1(C). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kleinian group」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.