Let $G$ be $SL_2({\mathbb C})$ and for $a,b\in G$ let $[a,b]=aba^{-1}b^{-1}$ be the commutator bracket. Let $n$ be a natural number $\ge 2$ and let $X\subset G^{2n}$ be the set of all $g\in G^{2n}$ such that $$ [g_1,g_2]\cdots[g_{2n-1},g_{2n}]=1. $$ The first question is, whether $X$ is connected. If not, can one give a list of the connected components? Finally, does the subset $SU(2)^{2n}\cap X$ meet every connected component?

If the last question has an affirmative answer, every $G$-valued representation of the fundamental group $\Gamma$ of a compact Riemann surface of genus $n$ can be deformed to a unitary one, which explains the title of my question.