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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!

Groupoid cardinality

Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.

There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.