The Moduli of Flat U(p,1) Structures on Riemann Surfaces

For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(\pi_1(X), U(p,1))/U(p,1)$ is the moduli space of flat $\U(p,1)$-connections on $X$. There is an integer invariant, $\tau$, the Toledo invariant associated with each element in $\Hom(\pi_1(X), U(p,1))/U(p,1)$. If $q = 1$, then $-2(g-1) \le \tau \le 2(g-1)$. This paper shows that $\Hom(\pi_1(X), U(p,1))/U(p,1)$ has one connected component corresponding to each $\tau \in 2Z$ with $-2(g-1) \le \tau \le 2(g-1)$. Therefore the total number of connected components is $2(g-1) + 1$.