Display variety

In mathematics , representation varieties are an important tool in group theory , topology, and geometry .

definition

Let be a finitely generated group and a semisimple algebraic Lie group . Be a presentation of a finite number of producers. One gives ${\displaystyle \Gamma }$ ${\displaystyle G}$ ${\displaystyle \Gamma =\langle x_{1},\ldots ,x_{n}\mid R_{1},\ldots ,R_{r},\ldots \rangle }$ ${\displaystyle \Gamma }$

${\displaystyle \operatorname {Hom} (\Gamma ,G)=\left\{f\colon \Gamma \rightarrow G{\text{ Homomorphismus}}\right\}}$

the structure of an algebraic set (i.e. a not necessarily irreducible algebraic variety ) as

${\displaystyle \operatorname {Hom} (\Gamma ,G)=\left\{(g_{1},\ldots ,g_{n})\in G^{n}\mid R_{1}(g_{1},\ldots ,g_{n})=e,\ldots ,R_{r}(g_{1},\ldots ,g_{n})=e,\ldots ,\right\}}$,

where the neutral element of denotes and a homomorphism is identified with the tuple . ${\displaystyle e}$ ${\displaystyle G}$ ${\displaystyle f\colon \Gamma \rightarrow G}$ ${\displaystyle (g_{1},\ldots ,g_{n})=(f(x_{1}),\ldots ,f(x_{n}))}$

It follows from Hilbert's basic theorem that the variety can already be defined by a finite number of equations.

It can be shown that the isomorphism type of the representation variety does not depend on the selected generating system.

Formation of quotients

The group acts through conjugation ${\displaystyle G}$ ${\displaystyle \operatorname {Hom} (\Gamma ,G)}$

${\displaystyle (gf)(\gamma )=gf(\gamma )g^{-1}}$.

The quotient is generally not a variety, instead one often looks at a somewhat smaller quotient, the character variety . ${\displaystyle \operatorname {Hom} (\Gamma ,G)/G:=\operatorname {Hom} (\Gamma ,G)/{\text{conj.}}}$ ${\displaystyle {\mathcal {R}}(\Gamma ,G)=\operatorname {Hom} (\Gamma ,G)//G}$

Non-algebraic groups

For arbitrary (not necessarily algebraic) semi-simple Lie groups one can define the representation variety as an analytic manifold as above . The quotient space is generally not Hausdorffian . However has ${\displaystyle \operatorname {Hom} (\Gamma ,G)/G:=\operatorname {Hom} (\Gamma ,G)/conj.}$

${\displaystyle \operatorname {Hom} ^{red}(\Gamma ,G):=\left\{\rho \in \operatorname {Hom} (\Gamma ,G):\not \exists P{\text{ parabolisch mit }}\rho (\Gamma )\subset P\right\}}$

(i.e. the submanifold of those homomorphisms whose picture is not in a parabolic subgroup ) an analytic manifold as a quotient.

Invariants

Let it be a CW complex with a fundamental group , then each representation corresponds to a flat bundle ${\displaystyle X}$ ${\displaystyle \pi _{1}X=\Gamma }$ ${\displaystyle \rho \colon \Gamma \to G}$

${\displaystyle G\to E_{\rho }\to X}$

and the topological invariants of the bundles defined by means of obstruction theory are invariants of the connected components of the representation variety.

The first class of obstruction

${\displaystyle o_{1}\in H^{1}(X,\pi _{0}(G))}$

disappears when is contiguous . ${\displaystyle G}$

The second class of obstruction

${\displaystyle o_{2}\in H^{2}(X,\pi _{1}(G))}$

corresponds for the Euler class and for the second Stiefel-Whitney class . ${\displaystyle G=SL(2,\mathbb {R} )}$ ${\displaystyle G=SL(n,\mathbb {R} ),n\geq 3}$ ${\displaystyle w_{2}(E_{\rho })}$

literature

• Alexander Lubotzky , Andy Magid: Varieties of representations of finitely generated groups . In: Mem. Amer. Math. Soc. , 58, 1985, no.336.
• Michail Kapovich : Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Boston MA 2009, ISBN 978-0-8176-4912-8
• Marc Culler , Peter Shalen : Varieties of group representations and splittings of 3-manifolds . In: Ann. of Math. , (2) 117, 1983, no. 1, pp. 109-146, doi: 10.2307 / 2006973 , JSTOR 2006973 .
• William Goldman : Topological components of spaces of representations . In: Invent. Math. , 93, 1988, no. 3, pp. 557-607; doi : 10.1007 / BF01410200