October 6,
David Fisher, Indiana University
First cohomology and local rigidity
Abstract:
In 1964, Andre Weil showed that a homomorphism $\pi$ from
a finitely generated group $\Gamma$ to a Lie group $G$ is
{\em locally rigid} whenever $H^1(\G,\mathfrak g)=0$.
Here $\pi$ is locally rigid if any nearby homomorphism is
conjugate to $\pi$ by a small element of $G$, and
$\mathfrak g$ is the Lie algebra of $G$. The point of my
talk is describe an infinite dimensional analog of Weil's
theorem:
\begin{theorem}Let $\Gamma$ be a finitely presented group,
$(M,g)$ a compact Riemannian manifold and
$\pi:\G{\rightarrow}\Isom(M,g){\subset}\Diff^{\infty}(M)$
a homomorphism. Then if $H^1(\G,\Vect^{\infty}(M))=0$, the
homomorphism $\pi$ is locally rigid as a homomorphism into
$\Diff^{\infty}(M)$. \end{theorem}
This work is motivated by a string of results by many
authors, beginning with Zimmer, generalizing results about
finite dimensional representations of lattices in Lie
groups to representations taking values in diffeomorphism
groups.
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