An improvement on a theorem of Ben Martin
Amnon Neeman
Abstract.
Let $\pi$ be the fundamental group of a Riemann surface of
genus $g \ge 2$. The group $\pi$ has a well known presentation,
as the quotient of a free group on generators $\{a_1,a_2,\dots,
a_g,b_1,b_2,\dots,b_g\}$ by the one relation
\[ [a_1,b_1][a_2,b_2]\dots[a_g,b_g] = 1. \]
This gives two inclusions $F \hookrightarrow \pi$, where $F$ is
the free group on $n$ 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 $\pi
\rightarrow G$ restricts to two representations $f_1: F_1
\rightarrow G$ and $f_2: F_2 \rightarrow G$. We prove that
on a Zariski open, dense subset of the space of pairs of
representations $\{f_1,f_2\}$, there exists a representation
$f: \pi \rightarrow G$ 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.