@SanathDevalapurkar And how do you propose to use dessin de'nfants?

To my understanding, G-T theory is no use here.

and how do you propose to "relate dessin d'enfants to this inverse limit construction"?

there's no apparent connection. do you have something else in mind?

@SanathDevalapurkar

I kind of have something of an idea, if you must know, @Sanath, but it's not entirely interesting.

Probably irrelevant

@Sanath digression : if you have an algebraic variety X over $\bar {\mathbf Q}$, and you have the corresponding dessin $\Gamma_X$, shouldn't $\text{Gal}(\mathbf Q(X)/\mathbf Q)$ act on $\Gamma_X$ by isometries?

was just wondering, because if it does, then this galois group must also act on the collection of little nbhds around the nodes of $\Gamma_X$ on the corresponding $\{0, 1, \infty\}$-branched Riemann surface transitively, so one should be able to realize this thing as a subgroup of automorphisms of the Riemann surface.

and then one should be able to tell something about the order of this, I suppose, by Hurwitz's theorem.

Hello @Sanath

Oh. I thought this was the Homotopy Theory room that you mentioned in the regular chat. I will leave

I have just used Belyi's theorem right up there

there would be no $\Gamma_X$ if there was no Belyi.