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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 Bianchi group ： ウィキペディア英語版 Bianchi group In mathematics, a Bianchi group is a group of the form :$PSL\_2(\backslash mathcal\_d)$ where ''d'' is a positive square-free integer. Here, PSL denotes the projective special linear group and $\backslash mathcal\_d$ is the ring of integers of the imaginary quadratic field $\backslash mathbb(\backslash sqrt)$. The groups were first studied by as a natural class of discrete subgroups of $PSL\_2(\backslash mathbb)$, now termed Kleinian groups. As a subgroup of $PSL\_2(\backslash mathbb)$, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space $\backslash mathbb^3$. The quotient space $M\_d\; =\; PSL\_2(\backslash mathcal\_d)\; \backslash backslash\backslash mathbb^3$ is a non-compact, hyperbolic 3-fold with finite volume, which is also called ''Bianchi manifold''. An exact formula for the volume, in terms of the Dedekind zeta function of the base field $\backslash mathbb(\backslash sqrt)$, was computed by Humbert as follows. Let $D$ be the discriminant of $\backslash mathbb(\backslash sqrt)$, and $\backslash Gamma=SL\_2(\backslash mathcal\_d)$, the discontinuous action on $\backslash mathcal$, then :$vol(\backslash Gamma\backslash backslash\backslash mathbb)=\backslash frac\}\}\backslash zeta\_)\}(2)\; \backslash \; .$ The set of cusps of $M\_d$ is in bijection with the class group of $\backslash mathbb(\backslash sqrt)$. It is well known that any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.〔Maclachlan & Reid (2003) p.58〕 ==References== * * * * * 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「Bianchi 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.