CHARACTER VARIETY OF REPRESENTATIONS OF A FINITELY GENERATED GROUP IN SL2
CHARACTER VARIETY OF REPRESENTATIONS OF A FINITELY GENERATED GROUP IN SL2
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SL2 中有限生成群表示的特征多样性
DOI：
10.1142/9789814503921_0014
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
Kyoji Saito
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文献类型：
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作者：
Kyoji Saito
本文是[S4，S5，S6]的一部分，[S4，S5，S6在SL 2和GL 2中的（n-生成的）群Γ，试图在几何中应用：Teichmüller空间，纽结理论，双曲流形，模空间，· · ·，等等。（参见例如[A]、[Be]、[Br]、[C-C-G-L-S]、[C-S]、[F-K]、[G]、[H]、[H-L-M]、[H-K]、[J-W]、[K]、[Kj]、[Ko]、[Kr]、[Ma]、[Mu]、[N-Z]、[O]、[S]、[S-S]、[T]、[W]、[We]，[Wo]、[Y]、· · ·等）。为了简单起见，我们在本论述中省略了GL 2的情况。让我们解释一下本文的主要结果。设r是一个群。本文引入特征标簇Ch（Γ，SL 2），以函子的方式参数化Γ在SL 2中的表示的共轭类.首先，引入的字符簇作为一个计划在Z上，独立的系数环的表示问题。然后，标度场被专门化到R，以获得关于经典拓扑的SL 2（R）和SU（2）中的表示的结果。让我们简单地解释一下。设Hom（Γ，SLn）是函子R ∈ {有1} 7的交换环→ Hom（Γ，SLn（R））∈ {sets}.函子是可表示的（见§1.3引理），因此，为了避免符号滥用，我们用相同的Hom（Γ，SLn）表示Z上表示函子的概型。群方案PGLn作用于Hom（Γ，SLn）。定义在Z上的泛范畴商Hom（Γ，SLn）//PGLn（Mumford [Mu 1]）是否存在似乎是一个难以解决的问题。不是直接要求商空间，我们引入i）特征簇Ch（Γ，SL 2）及其判别子簇DΓ作为Z抽象上的schemes，和ii）PGL 2不变态射πΓ：Hom（r，SL2）-Ch（r，SL2），证明了：i）πΓ在π −1 Γ（DΓ）的补上的限制是关于埃塔尔拓扑的主PGL 2-丛，ii）逆象π Γ（DΓ）是Hom（Γ，SL 2）的一个由交换或可约表示组成的子函子.这意味着补Hom（Γ，SL 2）：= Hom（Γ，SL 2）\π（DΓ）由绝对不可约表示组成，并且Hom（Γ，SL 2）具有定义在Z上的泛范畴商空间Ch（Γ，SL 2）：= Ch（Γ，SL 2）\DΓ。那么结果就是特化的
This is a partial exposition of [S4, S5, S6], which study the space of representations of a (finitely generated) group Γ in SL2 and GL2 in an attempt of its application in geometry: Teichmüller spaces, knot theory, hyperbolic manifolds, moduli spaces, · · · , etc. (see for instance, [A], [Be], [Br], [C-C-G-L-S], [C-S], [F-K], [G], [H], [H-L-M], [H-K], [J-W], [K], [Kj], [Ko], [Kr], [Ma], [Mu], [N-Z], [O], [S], [S-S], [T], [W], [We], [Wo], [Y], · · · , etc). For simplicity, we omit the case for GL2 in the present exposition. Let us explain the main result of the present paper. Let Γ be a group. The purpose of the present paper is to introduce the character variety Ch(Γ, SL2) in order to parameterize conjugacy classes of representations of Γ in SL2 in a functorial way. At first, the character variety is introduced as a scheme over Z, independent of the coefficient ring of representations in question. Then, the scaler field is specialized to R to obtain results on representations in SL2(R) and in SU(2) with respect to the classical topology. Let us explain this briefly. Let Hom(Γ, SLn) be the functor R ∈ {commutative rings with 1} 7→ Hom(Γ, SLn(R)) ∈ {sets}. The functor is representable (see §1.3 Lemma) and so, for an abuse of notation, we denote by the same Hom(Γ, SLn) the scheme over Z representing the functor. The group scheme PGLn acts on Hom(Γ, SLn). Whether the universal categorical quotient Hom(Γ, SLn)//PGLn (Mumford [Mu1]) defined over Z exists or not seems to be a hard and unsolved question. Instead of asking directly for the quotient space, we introduce i) the character variety Ch(Γ, SL2) together with its discriminant subvariety DΓ as schemes over Z abstractly, and ii) the PGL2-invariant morphism πΓ : Hom(Γ, SL2) → Ch(Γ, SL2), for which we prove that i) the restriction of πΓ on the complement of π −1 Γ (DΓ) is a principal PGL2-bundle with respect to the etal topology, and ii) the inverse image π Γ (DΓ) is a subfunctor of Hom(Γ, SL2) consisting of abelian or reducible representations. This implies that the complement Hom(Γ, SL2) := Hom(Γ, SL2)\π (DΓ) consists of absolutely irreducible representations, and that Hom(Γ, SL2) has the universal categorical quotient space Ch(Γ, SL2) := Ch(Γ, SL2)\DΓ defined over Z. Then the result is specialized