# What is the virtue of profinite groups as mathematical objects?

In my own research I use profinite groups quite frequently (for Galois groups and etale fundamental groups). However my use of them amounts to book-keeping: I only care about finite levels (finite Galois extensions; finite covers) and so I take their inverse limit. Then various "topological" arguments just mean: look at the finite levels (this occurs because the quotients by the open subgroups exactly correspond to the finite levels I care about.)

There are people who use profinite groups more intently than I do, and I suspect that they see some value in them as mathematical objects. So my question is this:

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What theorems/properties are good about profinite groups that don't arise trivially from its book-keeping nature (so for example the profinite Sylow theorems are disqualified, because they arise trivially from the finite-level group theory). What, if anything, rewards us for dealing with this new type of object rather than with the finite levels? (again, except that it makes it easier to write down notes.)

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You may like this paper: math.virginia.edu/~lls2l/are_all_groups.pdf – Kimball Dec 16 '10 at 4:43
This question is very closely related to mathoverflow.net/questions/49483/… – Theo Johnson-Freyd Dec 16 '10 at 5:21
This paper might be relevant to your question: front.math.ucdavis.edu/0604.5399 – Ian Agol Jun 27 '11 at 15:33

More generally, you might ask "what is the point of constructing limits or colimits of diagrams of objects instead of working directly with the diagrams?" A generic answer is that a diagram of objects in a category describes a functor, and it is useful to know that that functor is representable. For example, if objects $X_1, X_2$ in a category have a product $X_1 \times X_2$, this is equivalent to the statement that the functor $\text{Hom}(-, X_1) \times \text{Hom}(-, X_2)$ is representable. So you now know that this functor takes colimits to limits, which is new information.

Another generic answer is the following. Any time you construct an object $X$ as a limit of a diagram of other objects $X_i$ in a category, you know what the maps into $X$ look like by definition (compatible systems of maps into the $X_i$). What you don't know is what the maps out of $X$ look like, and this is new information you get from the existence of $X$. For example, the limit of the empty diagram is the terminal object $1$, and while maps into $1$ are trivial, maps out of $1$ ("global points") are not; in the category of schemes over a field $k$, for example (an example within an example!), they correspond to $k$-points.

Specializing to Galois theory, when you construct a Galois group $G$ as a limit of finite Galois groups $G_i$, the new information you have access to is, for example, the representation theory of $G$. I don't know how one would talk about the correspondence between modular forms and 2-dimensional Galois representations without direct access to the group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, for example.

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There are certainly situations where it suffices to consider the diagram of finite groups, e.g., the image of any continuous map from a profinite group to $GL_n(\mathbb{C})$ is finite. If you are only concerned with such families of representations, you don't have to worry about profinite groups.

On the other hand, there are situations where we want to look at continuous representations (e.g., of an profinite Galois group) on vector spaces with finer topology (e.g., $\ell$-adic or $p$-adic groups) and these often have infinite image. In principle, it is still possible to view these as compatible systems of representations on groups like $GL_n(\mathbb{Z}/\ell^k\mathbb{Z})$ for $k \geq 1$, but to me, it seems easier to consider a single continuous map and let the topology do its job.

Similarly, we may want to look at Galois representations that are slightly discontinuous (e.g., where the image of some Frobenius has infinite order) - one typically fixes this discontinuity by choosing a subgroup of the Galois group where Frobenius lifts don't generate compact groups. If we try to view this as a compatible system of finite Galois representations, it seems to be somewhat cumbersome.

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A uniform pro-$p$ group is an inverse limit of finite $p$-groups, a Lie group, and (the restriction to $\mathbb{Z}_p$ of) a finite-dimensional Lie algebra all at the same time. I think you'd be hard-pressed to see all these connections just by looking at a system of finite groups.

There are a lot of results in profinite group theory that say "$G$ has a subgroup of finite index such that..." without necessarily giving any way to bound the index. You can't see these properties by looking at individual finite images.

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Gildenhuys, Ribes and Zakesskii and others have developed a Bass-Serre theory for profinite groups acting on profinite trees. Using this theory, Ribes and Zalesskii showed that if $H_1,\ldots,H_n$ are finitely generated subgroups of a free group then the subset $H_1\cdots H_n$ is closed in the profinite topology. This was a conjecture by people working in semigroup theory and automata theory that is essentially equivalent to a conjecture of Rhodes in semigroup theory. The Ribes and Zalesskii proof does not use approximation by finite groups, but rather the geometry of these profinite trees. Other proofs using geometric group theory and using model theory now exist.

Recently Almeida assigned profinite groups to irreducible symbolic systems in a way that is functorial up to inner automorphism. Again finite groups do not explicitly appear in the discussion.

UPDATE: Another aspect of the theory of profinite groups as objects in their own right is the study of just infinite profinite groups. An infinite profinite group is just infinite if all its non-trivial closed normal subgroups are open. For example, the p-adic integers are just infinite. Just infinite is the analogue of simple for infinite profinite groups. Every finitely generated infinite profinite group has a just infinite quotient. There is a trichotomy due to Wilson (and refined by Grigorchuk) describing what they can look like. The study of just infinite profinite groups is connected to the theory of profinite branch groups and actions on rooted trees. See the handbook chapter by Bartholdi, Grigorchuk and Sunik.

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"Every finitely generated infinite profinite group has a just infinite quotient." Finite generation is not quite the right criterion (it is for pro-$p$ groups and for abstract groups, but for different reasons). See this paper for details: Zalesskii, P. A., Profinite groups admitting just infinite quotients. Monatsh. Math. 135 (2002), no. 2, 167--171. – Colin Reid Jul 9 '11 at 13:29
Thanks for the correction. I had in mind the argument for abstract groups, but it doesn't work in the profinite setting because one has to take closure. – Benjamin Steinberg Jul 9 '11 at 20:49