An improvement on a theorem of Ben Martin

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 \[ [a_1,b_1][a_2,b_2]\cdots[a_g,b_g]=1. \] 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 $\pi\longrightarrow G$ restricts to two representations $f_1:F_1\longrightarrow G$, $f_2:F_2\longrightarrow G$. We prove that on a Zariski open, dense subset of the space of pairs of representations $\{f_1,f_2\}$, there exists a representation $f:\pi\longrightarrow G$ 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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Published 1 January 2000