Hello.

right.

clears throat

actually the thing is that i have no idea what the solenoid could possibly me, though i have some prototypes.

ok, ok, let's just begin from the beginning.

consider the $\mathbf Z_p$. that is, the $p$-adic numbers.

this is the simplest kind of infinite profinite group one can think of. now, we want a geometric visualization of Z_p.

i.e., something on which Z_p acts geometrically.

the best way to do it is to look for action of Z/p^nZ on a class of geometric objects and then take the inverse limit.

the thing that comes to mind is the riemann surface of w = z^(p^n). the monodromy of this fellow is Z/p^nZ, so there is an obvious action.

so take the collection of all riemann surfaces of w = z^(p^i) over C\{0} (galois away from branch points, so chuck out branch point) and take the inverse limit.

one expects that Z_p acts on this fellow "nicely"

we'll show that it does.

first, note that since you've already chucked out {0}, each of the riemann surfaces deformation retracts onto S^1. so this space of ours is really homotopy equivalent to inverse limit of S^1's, with bonding maps z \to z^{p^n}

this is our solenoid.

? we haven't really begun proving anything yet

ok?

so, claim : the solenoid is a Z_p-bundle over S^1

to prove it, note that there is a diagram R \to \lim S^1 \leftarrow Z_p, the first map being induced by the universality of inverse limits, the second map from the inclusion Z/p^nZ \hookrightarrow R/p^nZ

take the pullback of this diagram to conclude the proof.

now it just becomes obvious that Z_p is the monodromy of the projection map onto S^1. so there you have something like a good action.

my goal is now to generalize this thing to realize galois groups geometrically.

one way to do this for Gal(\bar C(z)/C(z)) is to take inverse limit of all algebraic riemann surfaces over C, but i haven't really thought about this.

i have thought about a couple more interesting stuff related to this, but nothing too complete to talk about.

@SanathDevalapurkar yeah. and please don't write Z_(p) :P

it looks like localization of Z at (p) or something.

nothing fun. a few universal hyperbolic solenoids and associated monodromies. just a step forward at describing Gal(\bar C(z)/C(z))

so, you think you have anything that might contribute to this idea?

nope. i am not sure if that'd be the right kind of solenoid, actually.

actually, yeah, you can think of complex algebraic varities.

okey.

huh?

surely you mean a connected algebraic variety over \bar Q, not Q itself?

anyway, it doesn't matter. it's not "geometric enough"

the only thing relevant here is grothendieck teichmuller theory, but it's not related to this approach.

if such a solenoid is found, it'll have some implication towards galois (co)homology, which i'm sure is obvious to you.

[it's because such a space is K(gal(\bar q/q), 1), homology of which is precisely gal hmlgy]

bubyes