Galois action on compact hyperbolic surfaces and their associated solenoids.

Let S be a compact hyperbolic surface uniformised by a Fuchsian group
Γ. For any element σ ∈ G:= Gal(C/Q) the natural Galois action of G
on the coefficients of the algebraic equation corresponding to the Riemann
surface S yields a new hyperbolic surface S σ with uniformising group Γ σ .
Little seems to be known about the relationship between Γ and Γ σ as
subgroups of P SL 2 (R). I will attempt to show that by studying the action of
G on the solenoid associated to S one can find some invariants of this Galois
action. I will apply these results to present explicit (arithmetic) Fuchsian
groups that uniformise non-Galois-conjugate hyperbolic surfaces.