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In this work, we consider relative character varieties for representations ofthe 3-punctured sphere group in PU(2,1). We provide necessary and sufficientconditions on the peripheral conjugacy classes, for such a representation toadmit a decomposition as products of special elliptic elements. We prove thatthe representations satisfying these conditions form fibers of the so-calledmomentum map for PU(2,1). We apply these results to representations of the evensubgroup of triangle groups, and describe components of the associatedcharacter variety.

Triangle groups in the complex hyperbolic plane and special fibers of the momentum map in PU(2,1)