The Barbot component of discrete faithful representations of the modular group into Isom(SL_3(R)/SO(3)) is homeomorphic to R² × [0,∞), with Pappus representations on the boundary and Anosov representations in the interior.
Putting everything together, we see that ( c1,d 1) and (c2,d 2) give representations that are conjugate in Isom( X) only if they lie in the same θ4-orbit
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On Pappus and Anosov Representations of the Modular Group
The Barbot component of discrete faithful representations of the modular group into Isom(SL_3(R)/SO(3)) is homeomorphic to R² × [0,∞), with Pappus representations on the boundary and Anosov representations in the interior.