Posts Tagged ‘groupoids’

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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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.


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My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.


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