Abstract

For a compact Riemann surface X of genus g > 1, Hom(π₁(X),PU(p,q)) ∕ PU(p,q) is the moduli space of flat PU(p,q)-connections on X. There are two integer invariants, d_P,d_Q, associated with each σ in Hom(π₁(X),PU(p,q)) ∕ PU(p,q). These invariants are related to the Toledo invariant τ by τ = 2. This paper shows, via the theory of Higgs bundles, that if q = 1, then −2(g − 1) ≤ τ ≤ 2(g − 1). Moreover, Hom(π₁(X),PU(2,1)) ∕ PU(2,1) has one connected component corresponding to each τ in Z with −2(g − 1) ≤ τ ≤ 2(g − 1). Therefore the total number of connected components is 6(g − 1) + 1.

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