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Research Report MRR99-027
An improvement on a theorem of Ben Martin
Amnon Neeman
Abstract:
Let $\pi$ be the fundamental group of a Riemann surface
of genus $g\geq2$. The group $\pi$ has a well--known presentation,
as the quotient of a free group on generators
$\{a_1,a_2,\ldots,a_g,b_1,b_2,\ldots,b_g\}$ by the one relation
\bigbreak
\centerline{$[a_1,b_1][a_2,b_2]\cdots[a_g,b_g]=1$.}
\bigbreak
This gives two inclusions $F\hookrightarrow \pi$, where
$F$ is the free group on $g$ generators; we could map the
generators to the $a$'s, or to the $b$'s. Call the images of these
inclusions $F_1\subset\pi$ and $F_2\subset\pi$.
Given a connected, reductive group $G$ over an algebraically
closed field of characteristic 0, any representation $\pi\to G$
restricts to two representations $f_1:F_1\to G$, $f_2:F_2\to G$.
We prove that on a Zariski open, dense subset of the space of
pairs of representations $\{f_1,f_2\}$, there exists a
representation $f:\pi\to G$ lifting them, up to (separate)
conjugacy of $f_1$ and $f_2$.
Ben Martin proved this theorem, with the hypothesis that the
semisimple rank of $G$ is $> g$. We remove the hypothesis.
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