Dynamics of the Mapping Class Group on the Moduli of a Punctured Sphere with Rational Holonomy

Let M be a four-holed sphere and Γ the mapping class group of M fixing the boundary ∂M . The group Γ acts on MB(SL(2,C)) = Hom+B (π1(M), SL(2,C))/ SL(2,C) which is the space of completely reducible SL(2,C)-gauge equivalence classes of flat SL(2,C)-connections on M with fixed holonomy B on ∂M . Let B ∈ (−2, 2) and MB be the compact component of the real points of MB(SL(2,C)). These points correspond to SU(2)representations or SL(2,R)-representations. The Γ-action preserves MB and we study the topological dynamics of the Γ-action onMB and show that for a dense set of holonomy B ∈ (−2, 2), the Γorbits are dense in MB. We also produce a class of representations ρ ∈ Hom B (π1(M), SL(2,R)) such that the Γ-orbit of [ρ] is finite in the compact component of MB(SL(2,R)), but ρ(π1(M)) is dense in SL(2,R).