For X a smooth projective curve over C of genus g > 1, Hom +(π1(X), U(p, 1))/ U(p, 1) is the moduli space of flat semi-simple U(p, 1)-connections on X. There is an integer invariant, τ, the Toledo invariant associated with each element in Hom+(π 1(X), U(p, 1))/ U(p, 1). This paper shows that Hom +(π1(X), U(p, 1))/U(p, 1) has one connected component corresponding to each τ ε 2ℤ with -2(g - 1) ≤ τ ≤ 2(g - 1). Therefore the total number of connected components is 2(g - 1) + 1.
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