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## Conjugacy classes of finite index subgroups

Previously we learned how to count the number of finite index subgroups of a finitely generated group $G$. But for various purposes we might instead want to count conjugacy classes of finite index subgroups, e.g. if we wanted to count isomorphism classes of connected covers of a connected space with fundamental group $Gi$.

There is also a generating function we can write down that addresses this question, although it gives the answer less directly. It can be derived starting from the following construction. If $X$ is a groupoid, then $LX = [S^1, LX]$, the free loop space or inertia groupoid of $X$, is the groupoid of maps $S^1 \to X$, where $S^1$ is the groupoid $B\mathbb{Z}$ with one object and automorphism group $\mathbb{Z}$. Explicitly, this groupoid has

• objects given by automorphisms $f : x \to x$ of the objects $x \in X$, and
• morphisms $(f_1 : x_1 \to x_1) \to (f_2 : x_2 \to x_2)$ given by morphisms $g : x_1 \to x_2$ in $X$ such that

$x_1 \xrightarrow{f_1} x_1 \xrightarrow{g} x_2 = x_1 \xrightarrow{g} x_2 \xrightarrow{f_2} x_2$.

It’s not hard to see that $L(X \coprod Y) \cong LX \coprod LY$, so to understand this construction for arbitrary groupoids it’s enough to understand it for connected groupoids, or (up to equivalence) for groupoids $X = BG$ with a single object and automorphism group $G$. In this case, $LBG$ is the groupoid with objects the elements of $G$ and morphisms given by conjugation by elements of $G$; equivalently, it is the homotopy quotient or action groupoid of the action of $G$ on itself by conjugation.

In particular, when $G$ is finite, this quotient always has groupoid cardinality $1$. Hence:

Observation: If $X$ is an essentially finite groupoid (equivalent to a groupoid with finitely many objects and morphisms), then the groupoid cardinality of $LX$ is the number of isomorphism classes of objects in $X$.

I promise this is relevant to counting subgroups!

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## Forms and Galois cohomology

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

$\displaystyle C(k) \cong C(L)^G$

where $f : k \to L$ is a Galois extension, $G = \text{Aut}(L)$ is the Galois group of $L$ (thinking of $L$ as an object of the category of field extensions of $k$ at all times), $C(k)$ is a category of “objects over $k$,” and $C(l)$ is a category of “objects over $L$.”

In fact this description is probably only correct if $k \to L$ is a finite Galois extension; if $k \to L$ is infinite it should probably be modified by requiring that every function of $G$ that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on $G$. To avoid this difficulty we’ll stick to the case that $k \to L$ is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that $k$-forms $c_k \in C(k)$ of an object $c_L \in C(L)$ can be classified by Galois cohomology $H^1(BG, \text{Aut}(c_L))$, and we’ll give some examples.

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## Stating Galois descent

After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent.

Fix a field $k$. The gadgets we want to study assign to each separable extension $k \to L$ a category $C(L)$ of “objects over $L$,” to each morphism $f : L_1 \to L_2$ of extensions an “extension of scalars” functor $f_{\ast} : C(L_1) \to C(L_2)$, and to each composable pair $L_1 \xrightarrow{f} L_2 \xrightarrow{g} L_3$ of morphisms of extensions a natural isomorphism

$\displaystyle \eta(f, g) : f_{\ast} g_{\ast} \cong (fg)_{\ast}$

of functors $C(L_1) \to C(L_3)$ (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms $f_{\ast} g_{\ast} h_{\ast} \cong (fgh)_{\ast}$ we can write down from this data agree. We’ll also want unit isomorphisms $\varepsilon : \text{id}_{C(L)} \cong (\text{id}_L)_{\ast}$ satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of $k$ to the 2-category $\text{Cat}$, and in particular each $C(L)$ naturally acquires an action of $\text{Aut}(L)$ (where we mean automorphisms of extensions of $k$, hence if $L$ is Galois this is the Galois group) in precisely the sense we described earlier.

We’ll call such an object a Galois prestack (of categories, over $k$) for short. The basic example is the Galois prestack of vector spaces $\text{Mod}(-)$, which sends an extension $L$ to the category $\text{Mod}(L)$ of $L$-vector spaces and sends a morphism $f : L_1 \to L_2$ to the extension of scalars functor

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

Every example we consider will in some sense be an elaboration on this example in that it will ultimately be built out of vector spaces with extra structure, e.g. the Galois prestacks of commutative algebras, associative algebras, Lie algebras, and even schemes. In these examples, fields are not really the natural level of generality, and to make contact with algebraic geometry we should replace them with commutative rings, but for now we’ll ignore this.

In order to state the definition, we need to know that if $f : k \to L$ is an extension, then the functor $f_{\ast} : C(k) \to C(L)$ naturally factors through the category $C(L)^G$ of homotopy fixed points for the action of $G = \text{Aut}(L)$ on $C(L)$. We’ll elaborate on why this is in a moment.

Definition: A Galois prestack satisfies Galois descent, or is a Galois stack, if for every Galois extension $k \to L$ the natural functor $C(k) \to C(L)^G$ (where $G = \text{Aut}(L) = \text{Gal}(L/k)$) is an equivalence of categories.

In words, this condition says that the category of objects over $k$ is equivalent to the category of objects over $L$ equipped with homotopy fixed point structure for the action of the Galois group (or Galois descent data).

(Edit, 11/18/15:) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and its comments for some discussion.

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## Finite index subgroups of the modular group

Two weeks ago we proved the following formula. Let $G$ be a finitely generated subgroup and let $a_n$ be the number of subgroups of $G$ of index $n$. Then

$\displaystyle \sum_{n \ge 0} \frac{|\text{Hom}(G, S_n)|}{n!} z^n = \exp \left( \sum_{n \ge 1} \frac{a_n}{n} z^n \right)$.

This identity reflects, in a way we made precise in the previous post, the decomposition of a finite $G$-set (the terms on the LHS) into a disjoint union of transitive $G$-sets (the terms on the RHS).

Noam Zeilberger commented on the previous post that he had seen results like this for more specific groups in the literature; in particular, Samuel Vidal describes a version of this analysis for $G = \Gamma = PSL_2(\mathbb{Z})$, the modular group. In this post we’ll use the above formula to compute the number of subgroups of index $n$ in $\Gamma$ using a computer algebra system that can manipulate power series. We’ll also say something about how to visualize these subgroups.

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## Projective representations are homotopy fixed points

Yesterday we described how a (finite-dimensional) projective representation $\rho : G \to PGL_n(k)$ of a group $G$ functorially gives rise to a $k$-linear action of $G$ on $\text{Mod}(M_n(k)) \cong \text{Mod}(k)$ such that the Schur class $s(\rho) \in H^2(BG, k^{\times})$ classifies this action.

Today we’ll go in the other direction. Given an action of $G$ on $\text{Mod}(k)$ explicitly described by a 2-cocycle $\eta \in Z^2(BG, k^{\times})$, we’ll recover the category of $\eta$-projective representations, or equivalently the category of modules over the twisted group algebra $k \rtimes_{\eta} G$, by taking the homotopy fixed points of this action. We’ll end with another puzzle.

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## Projective representations give categorical representations

Today we’ll resolve half the puzzle of why the cohomology group $H^2(BG, k^{\times})$ appears both when classifying projective representations of a group $G$ over a field $k$ and when classifying $k$-linear actions of $G$ on the category $\text{Mod}(k)$ of $k$-vector spaces by describing a functor from the former to the latter.

(There is a second half that goes in the other direction.)

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## Projective representations

Three days ago we stated the following puzzle: we can compute that isomorphism classes of $k$-linear actions of a group $G$ on the category $C = \text{Mod}(k)$ of vector spaces over a field $k$ correspond to elements of the cohomology group

$\displaystyle H^2(BG, k^{\times})$.

This is the same group that appears in the classification of projective representations $G \to PGL(V)$ of $G$ over $k$, and we asked whether this was a coincidence.

Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.

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