Dreaming of a GTometry
In this post, I suggest a weird idea, which may very well be naive or wellknown, but that I felt like recording. Can we see varieties equipped with an outer action of the GrothendieckTeichmüller group on their geometric étale fundamental group as actual geometric objects defined over “something small”, as if we were doing “algebraic geometry over \(\GT\)”?
outer galois actions
Consider an algebraic variety \(X\) over \(\Q\), geometrically connected and equipped with a geometric basepoint \(x \in X(\Qbar)\) which we never write explicitely. Denote by \(X_{\Qbar}\) the extension of scalars \(X \underset{\Spec \Q}{\times} \Spec \Qbar\), and denote by \(\pigeom(X)\) the étale fundamental group of \(X_{\Qbar}\). Denote also by \(G_{\Q}\) the absolute Galois group \(\mathrm{Gal}(\Qbar\mid\Q)\). We have an exact sequence:
\[1 \to \pigeom(X) \to \piet(X) \to G_{\Q} \to 1.\]Since \(\pigeom(X)\) is a normal subgroup of \(\piet(X)\), \(\piet(X)\) has an inner action on \(\pigeom(X)\) via conjugacy. Two lifts of a \(\sigma \in G_{\Q}\) into an element of \(\piet(X)\) act on \(\pigeom(X)\) similarly up to conjugacy by an element of \(\pigeom(X)\). Therefore, we have an outer action of \(G_{\Q}\) on the geometric fundamental group \(\pigeom(X)\) (which is the profinite completion of the topological fundamental group):
\[G_{\Q} \to \Out(\pigeom(X)) = \Out(\widehat{\pi_1(X(\C))}).\]Moreover, each \(\Q\)point of \(X\) induces a lift of this outer action to an actual action \(G_{\Q} \to \Aut(\pigeom(X))\), welldefined up to global conjugacy by an element of \(\pigeom(X)\).
outer GTactions
Now, for some varieties (most importantly, and quite tautologically, for the moduli stacks of curves \(\mathcal{M}_{g,n}\)), people have found ways to extend the outer action of \(G_{\Q}\) into an action of the GrothendieckTeichmüller group \(\GT\). This “extension property” looks a lot like a “descent result”, suggesting that these varieties are somehow defined over a “field” smaller^{1} than \(\mathbb{Q}\), whose absolute Galois group is \(\GT\)…
We can even guess what the “\(\GT\)points” should be (in good cases, guided by Grothendieck’s section conjectures): they are lifts of the outer \(\GT\)action into an action \(\GT \to \Aut(\pigeom(X))\), modulo conjugacy by an element of \(\pigeom(X)\)!
Let us be even more radical — anabeliangeometry like — and take as the definition of a \(\GT\)variety^{2} that it is an extension of \(\GT\) by a finitely generated profinite group. Morphisms have a good candidate definition too, inspired by anabelian conjectures for hyperbolic curves: if \(1 \to A \to B \to \GT \to 1\) and \(1 \to A' \to B' \to \GT \to 1\) are two extensions, a morphism between them is a morphism \(B \to B'\), which commutes with the surjections to \(\GT\), and considered modulo conjugacy by an element of \(A'\).
This principle gives rise to a category of “geometriclike” objects for any group \(G\), which I call \(G\)ometries (defined carefully below). A variety over \(\Q\) induces a \(G_{\Q}\)ometry, and the “\(\GT\)varieties” presented above are the \(\GT\)ometries.
first Gometrical constructions: points
Fix a profinite group \(G\). Let us write explicitly the definitions suggested in the last paragraphs:
Definition (\(G\)ometry): A \(G\)ometry \(X\) is a profinite group – denoted by \(\piet(X)\) – equipped with a continuous surjection \(\piet(X) \to G\) whose kernel – denoted by \(\pigeom(X)\) – is a finitely generated profinite group. The surjection \(\piet(X) \to G\) is called the structural morphism of \(X\).
Definition (morphism of \(G\)ometries): A morphism between two \(G\)ometries \(X\) and \(Y\) is the class, modulo conjugacy by an element of \(\pigeom(Y)\), of a group morphism \(\piet(X) \to \piet(Y)\) which commutes with the structural morphisms.
Definition (\(G\)points): The point \(\star\) is the \(G\)ometry corresponding to the identity morphism \(\id_G: G \to G\). A \(G\)point of a \(G\)ometry \(X\) is a morphism of \(G\)ometries \(\star \to X\). We denote by \(X(G)\) the set of \(G\)points of \(X\).
A \(G\)point is the equivalence class of the section of the structural morphism under the conjugacy action of \(\pigeom(X)\). It also corresponds to a lift of the outer action \(G \to \Out(\pigeom(X))\) into an actual action \(G \to \Aut(\pigeom(X))\). A \(G\)ometry which admits a \(G\)point is a split extension of \(G\), the section corresponding to the \(G\)point. In this case, \(\piet(X)\) is realized as a semidirect product \(\pigeom(X) \rtimes G\).
The funny game is to attempt doing geometry with \(G\)ometries – and especially constructing “usual” spaces using these purely grouptheoretic objects. For example:
Definition (extension of scalars): Let \(H\) be an open subgroup of \(G\). The extension of scalars \(X_H\) of \(X\) to \(H\) is the \(H\)ometry obtained by letting \(\piet(X_H)\) be the subgroup of \(\piet(X)\) consisting of elements which are mapped to elements of \(H\) by the structural morphism.
Note that \(\pigeom(X_H) = \pigeom(X)\), and the outer action of \(H\) on \(\pigeom(X)\) is the restriction of the outer action of \(G\).
Define the \(H\)points of \(X\) as the \(H\)points of \(X_H\), and let \(X(H) = X_H(H)\). If \(Y\) is a \(H\)ometry, we say that it is defined over \(G\) if \(Y\) is isomorphic to \(X_H\) for some \(G\)ometry \(X\).
Let \(X\) be a \(G\)ometry and \(H\) be a normal open subgroup of \(G\). Denote by \(\varphi : G \to \Aut(H)\) the morphism induced by conjugacy. Consider an element \(g \in G\) and an \(H\)point of \(X\), represented by a morphism \(x : H \to \Aut(\pigeom(X))\). Define \(g.x\) to be the class of \(x \circ \varphi(g)\) modulo conjugacy by an element of \(\pigeom(X)\), which is an \(H\)point. This is welldefined since replacing \(x\) by another representative \(\alpha^{1} x \alpha\) with \(\alpha \in \pigeom(X)\) leads to:
\[(\alpha^{1} x \alpha) \circ \varphi(g) = \alpha^{1} (x \circ \varphi(g)) \alpha\]which defines the same \(H\)point as \(x \circ \varphi(g)\).
Hence, we have defined an action of \(G\) on the set \(X(H)\) of \(H\)points of \(X\).
A quick sanity check:
Proposition: An \(H\)point of \(X\) which extends to a \(G\)point of \(X\) is invariant under the action of \(G\).
Proof: By hypothesis, we have a morphism \(\tilde x : G \to \Aut(\pigeom(X))\) whose restriction \(x : H \to \Aut(\pigeom(X))\) represents the \(H\)point in question. Let \(g \in G\). For all \(h \in H\) we have:
\[(x \circ \varphi(g))(h) = x (g^{1}hg) = \tilde x(g)^{1} x(h) \tilde x(g)\]and thus \(x \circ \varphi(g)\) defines the same \(H\)point as \(x\). ■
Let \(X(\bar G)\) denote the inductive limit, indexed by normal open subgroups \(H\) of \(G\), of the sets \(X(H)\). Then \(G\) acts on the set \(X(\bar G)\).
Gometrical covers
Let \(X\) be a \(G\)ometry. In what follows, “cover” always means “finite étale connected cover”:
Definition (geometric/algebraic cover): A geometric (resp. algebraic) cover of \(X\) is a conjugacy class of open subgroups of \(\pigeom(X)\) (resp. \(\piet(X)\)). These subgroups are called markings of the cover. A cover consisting of a single normal subgroup (i.e. there is a single marking) is said to be Galois.
The degree of a cover is the index of the corresponding subgroups. The Galois closure of a cover is the cover corresponding to the normal closure of any of its markings. The outer action of \(G\) on \(\pigeom(X)\) induces an action of \(G\) on geometric covers of \(X\).
Definition (geometrically connected): An algebraic cover of \(X\) is geometrically connected if its markings, when intersected with \(\pigeom(X)\), form a single conjuacy class of subgroups of \(\pigeom(X)\).
A geometrically connected algebraic cover of \(X\) induces, by intersection with \(\pigeom(X)\), a geometric cover of \(X\).
Definition (defined over \(G\)): An algebraic cover of \(X\) is defined over \(G\) if the composition of inclusion (of any marking) into \(\piet(X)\) with the structural morphism is surjective and its kernel is a profinite finitely generated group. A geometric cover of \(X\) is defined over \(G\) if it is obtained as the intersection with \(\pigeom(X)\) of a geometrically connected algebraic cover of \(X\) which is defined over \(G\).
Geometrically connected covers defined over \(G\) are themselves \(G\)ometries.
A quick sanity check:
Proposition: The group \(G\) acts trivially on geometric covers which are defined over \(G\).
Proof: Consider an open subgroup \(H\) of \(\piet(X)\), which defines a geometrically connected algebraic cover of \(X\) defined over \(G\). We have an exact sequence:
\[1 \to H \cap \pigeom(X) \to H \to G \to 1\]Take an element \(g \in G\) and an arbitrary lifting \(\bar g \in H\), and in particular \(\bar g \in \piet(X)\) is a lifting of \(g\) in \(\piet(X)\). The outer action of \(g\) on \(\pigeom(X)\), and subsequently the action of \(g\) on geometric covers of \(X\), is induced by conjugation by \(\bar g\). Since \(H \cap \pigeom(X)\) is normal in \(H\) and \(\bar g \in H\), conjugacy by \(\bar g\) leaves it unchanged, so \(g\) acts trivially on the corresponding geometric cover. ■
Let us mention that a rationally marked \(G\)ometry is an extension of \(G\) by a profinite finitely generated group equipped with a splitting (defined up to conjugacy by an element of \(\pigeom(X)\)), that a marked cover (geometric or algebraic) is an open subgroup of a fundamental group (instead of a conjugacy class of subgroups), and that a rationally marked cover is an open subgroup \(H\) of \(\piet(X)\) which surjects onto \(G\), equipped with a splitting \(G \to H\), defined up to conjugacy by an element of \(\ker(H \to G)\).
Question: Harbater’s theory of patching shows that extensions of \(\Q_p(T)\) can be patched, in particular this solves the inverse Galois problem over \(\Q_p(T)\). André has defined a group \(\GT_{p}\) which should be the “\(p\)adic doppelgänger of \(\GT\)” — in particular \(\GT_p \cap G_{\Q} = \mathrm{Gal}(\bar\Q_p\mid\Q_p)\). Assuming we can show that \(\mathbb{P}^1\) with \(n\) points removed is defined over \(\GT_p\) (which is a first question), can one prove a form of patching for branched covers of \(\mathbb{P}^1\) defined over \(\GT_p\)?
applications
F. Pop^{3} has characterized \(G_{\Q}\) as the largest group which has outer actions on the geometric étale fundamental groups of all regular quasiprojective \(\mathbb{Q}\)varieties, which are compatible with all \(\mathbb{Q}\)morphisms. It seems then that understanding how different \(\GT\) and \(G_{\Q}\) actually are (and whether they are, actually, different) mostly amounts to understanding how often the outer Galois action on \(\pigeom\) can be extended into a \(\GT\)action: each variety for which this is shown to be the case brings \(\GT\) “a little closer” to \(G_{\Q}\). (notably, Ihara and Matsumoto have proved such extension results for algebraic configuration spaces^{4})
One of the reasons why a geometry over \(\GT\) may be of use is because it transforms this question into a (virtually) geometrical question: we are mostly looking at the “"”fields””” of definition of \(G_{\Q}\)ometries, and trying to show that they are actually “defined over \(\GT\)”, i.e. that \(\GT\) has an outer action on the geometric fundamental group that extends the Galois action. This rephrases the question in a language more familiar to algebraic geometers. An understanding as to how far this idea of “synthetic”, “groupbased”, “schemefree” algebraic geometry can be pushed, and in particular an attempt to construct various varieties (moduli spaces in particular) directly in the language of \(G\)ometries (or a more sophisticated version of that idea), could be a useful tool to see how the methods usually used to compute fields of definition of varieties (notably, descent criteria) may admit generalizations, or philosophical equivalents, in this language.
Bonus question: What could be said of the whole étale homotopy type? Such a generalization is clearly needed if one wishes to have a more faithful view of the geometry of things that are not hyperbolic curves. See this.
footnotes

Wait — is this anyhow related to \(\F_1\)? Many candidate definitions of a \(\F_1\)scheme have been proposed. It would be interesting if any of these had any link with the existence of an outer \(\GT\)action on a geometric étale fundamental group. ↩

This notion may or may not be related to the objects Mochizuki calls “anabeloids” (see here), I am unsure about that. ↩

This was only published in 2019, although everyone apparently knew that Pop had proven the result since 2002. See also this and that. What about this and that? ↩

There is also impressive work by BleherChinburgLubotzky concerning the extension of Galois representations to \(\GT\)representations. Maybe representations could be seen as geometrical objects the way étale fundamental groups are in this post (à la FontaineMazur) and give another form of “geometry over \(\GT\)”. ↩