Because rigidity

In mathematics , Weil's rigidity theorem gives a computable condition for the local rigidity (i.e., non-deformability) of representations . It is important in various areas of mathematics, mostly in connection with representations of discrete groups .

Tangent space of the representation variety

Representations of a group in a Lie group can be understood as points of the representation variety. Deformations of a representation are then curves in the representation variety and thus correspond to tangential vectors of the representation variety. ${\displaystyle \Gamma }$ ${\displaystyle G}$

The Zariski tangent space of the representation variety corresponds to the 1-coccycles with values ​​in the adjoint representation :

${\displaystyle T_{\rho }\operatorname {Hom} (\Gamma ,G)=Z^{1}(\Gamma ,Ad\circ \rho )}$.

A tangential vector corresponds to a curve (with ) in and the associated 1-cocycle is given by ${\displaystyle \rho _{t}}$ ${\displaystyle \rho _{0}=\rho }$ ${\displaystyle \operatorname {Hom} (\Gamma ,G)}$ ${\displaystyle z\in Z^{1}(\Gamma ,G)}$

${\displaystyle z(\gamma )=\left.{\frac {d}{dt}}\right\vert _{t=0}\rho _{t}(\gamma )\rho _{0}^{-1}(\gamma )}$.

The deformations given by conjugation (with ) correspond exactly to the 1-Korand. ${\displaystyle \rho _{t}(\gamma )=g_{t}\rho (\gamma )g_{t}^{-1}}$ ${\displaystyle g_{0}=e}$

In particular is locally rigid if ${\displaystyle \rho \colon \Gamma \to G}$

${\displaystyle H^{1}(\Gamma ,Ad\circ \rho )=0}$

is. The reverse is generally not true, but it is true for semi- simple representations . ${\displaystyle \rho }$

Weil's law of rigidity

Let be a connected semisimple Lie group with no compact factor and an irreducible co-compact lattice . If is not locally isomorphic to SL (2, R) then is ${\displaystyle G}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$

${\displaystyle H^{1}(\Gamma ,Ad)=0}$.

In particular, inclusion is locally rigid. ${\displaystyle \iota \colon \Gamma \to G}$

See also

• Mostow rigidity

literature

• André Weil: On discrete subgroups of Lie groups. I: Ann. of Math. (2) 72, 369-384 (1960). pdf II: Ann. of Math. (2) 75, 578-602 (1962). pdf
• Yozô Matsushima, Shingo Murakami: On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds. Ann. of Math. (2) 78: 365-416 (1963). pdf
• Armand Borel: Cohomologie et rigidité d'espaces compacts localement symétriques. Seminaire Bourbaki, 16e année: 1963/64, Fasc. 2, Exposé 265.
• André Weil: Remarks on the cohomology of groups. Ann. of Math. (2) 80: 149-157 (1964). pdf
• MS Raghunathan: On the first cohomology of discrete subgroups of semisimple lie groups. Amer. J. Math. 87, 103-139 (1965). pdf
• MS Raghunathan: Discrete subgroups of Lie groups. Results of Mathematics and its Frontier Areas, Volume 68. Springer-Verlag, New York-Heidelberg, 1972.
• N. Bergeron, T. Gelander: A note on local rigidity. Geom. Dedicata 107: 111-131 (2004). pdf

Web links

• Besson: Calabi-Weil infinitesimal rigidity
• Fisher: Local rigidity of Group Actions: past, present, future
• Kapovich: Group cohomology and representation varieties