The last two pages of Appendix C Milnor's characteristic Classes gives an example of a flat bundle with non-zero Euler class. I have a question about the structure group of this bundle.
The example starts with a Riemann surface of genus larger than 2. Its fundamental group,G, is represented as a subgroup of PSL(2;R) and acts on the upper half plane as the group of covering transformations of the surface.
This action preserves the extended real axis and gives a circle bundle over the surface, E = HxRP^1/G -> H/G = S.
proof is given that it is isomorphic to the bundle of tangent directions to the surface.
This bundle has a 2 fold cover corresponding to the cohomology class in H^1(E;Z2) that maps into the Thom class of the associated 2 plane bundle under the connecting homomorphism. This class exists because the second Stiefel Whitney class of the bundle is zero (the surface has even Euler characteristic).
So the 2 fold cover is another circle bundle and each fiber circle in the original is covered by a fiber circle of the two fold cover.
Now it is claimed that the structure group of this two fold cover comes from a representation of G in SL(2:R). I think this means that it is the quotient HxRP^1/G where G now acts on the projective line as a subgroup of SL(2;R) rather than PSL(2;R). This I do not understand.