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Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain.

Magnus, in his 1968 survey article, motivates residually finite groups by saying that residual finiteness allow us to extract information about the group in an algebraic manner. I understand and agree with this, and that was a fine motivation during the golden age of group theory. However, what about in today's world? How can we apply this property of groups to other settings?

So, I have two concrete questions.

  1. Why do we care whether hyperbolic groups are residually finite or not - we have soluble word problem, soluble isomorphism problem, Hofian, and so on. These properties arguably imply that Magnus' motivation does not hold. I should say that "because we don't know and it is an interesting question" is not really the answer I am looking for...(EDIT: I am aware that this implies that fundamental groups of hyperbolic $3$-manifolds are LERF, but, in a certain sense, this is still a group-theoretic property.)

  2. What are examples of theorems which say "this group is residually finite and therefore that amazing theorem in number theory holds!", or "this class of groups are residually finite so that class of rings have this wonderful property". That is, how does residual finiteness fit in to the big picture?

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    $\begingroup$ The next natural question would by why that theorem in number theory that holds is so amazing, or why the property of the class of rings is wonderful. $\endgroup$ – Alex Degtyarev Feb 26 '15 at 19:34
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    $\begingroup$ I think this is a fair question. Not all of us are group theorists, and even those of us who are might not spend much time pondering whether things are residually finite. (As a kind of harmonic analyst, I am interested in whether certain groups are residually finite, but that's a side-note) $\endgroup$ – Yemon Choi Feb 26 '15 at 19:37
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    $\begingroup$ Residual finiteness is a topological property. If roughly means that if $G=\pi_1(X)$ and $C$ is a compact subspace of the universal cover $\tilde X$ of $X$, then there is a finite cover $Y$ of $X$ such that the map $\tilde X\to Y$ is injective on $C$. (Here $X$ should be nice enough.) $\endgroup$ – Benjamin Steinberg Feb 26 '15 at 20:07
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    $\begingroup$ A proof that all hyperbolic groups are residually finite would be a major step forward in our understanding of the topology of hyperbolic spaces. Though for some reason, the 'big picture' for the OP only seems to involve algebra... $\endgroup$ – HJRW Feb 27 '15 at 1:33
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    $\begingroup$ I'm wondering why the anonymous user who asked the question claims that the late sixties were the golden age of group theory. It certainly was the golden age of the quest of finite simple groups but this is not the end of the story for finite groups, let alone infinite groups. I voted to close this question as primarily opinion-based (and found it essentially pointless), but it was reopened. It would be nice if people who found this question of mathematical interest could edit it so as to make it less polemical. $\endgroup$ – YCor Feb 27 '15 at 18:12
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A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called generalized Riemann Existence Theorem, whose proof is due to Grauert and Remmert, the finite coverings of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $\pi_1^{top}(X)$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

In Toledo's examples, $\ker \eta$ is free group of infinite rank. Later, F. Catanese and J. Kollar, and independently M. Nori, were able to find examples where $\ker \eta$ is a finite cyclic group. Such examples are described in

Classification of irregular varieties, Lecture Notes in Mathematics 1515, Springer 1992.

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  • $\begingroup$ Interesting answer. Judging on the popularity of the different subjects discussed on mathoverflow, we should perhaps rename it as AGflow. (Not to be confused with agflow.com) $\endgroup$ – Martin Brandenburg Aug 17 '15 at 9:33
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    $\begingroup$ In the early days, I thought that the only reason they didn't name it stackoverflow was that the name was already taken... $\endgroup$ – JBorger Aug 17 '15 at 12:14
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For some reason, the question seems to be asking for an algebraic (number- or ring-theoretical) justification for residual finiteness (and implicitly LERF, though in fact the correct statement is that every hyperbolic group is residually finite if and only if they are all QCERF, as proved by Agol--Groves--Manning). Recall that residual finiteness means that the trivial subgroup is separable, and QCERF means that all quasiconvex subgroups are separable.

But the principal motivation is topological (as Benjamin Steinberg has pointed out in comments)---this was why residual finiteness was key to the proof of the Virtual Haken Conjecture. Precisely, Scott observed that if $\Gamma=\pi_1X$ and $X$ is a cell complex, then a subgroup $H$ is separable if and only if, for any compact subet $K$ of the associated covering space $X^H\to X$, there is a factorization

$X^H\to X'\to X$

where $X'\to X$ is finite-sheeted and $K$ embeds into $X'$.

In other words, QCERF would enable us to promote immersions to embeddings in a finite-sheeted covering space. As a I said in comments, this would be a huge step forward in our understanding of the topology of such spaces $X$. There are lots of applications: the paper by Friedl and Vidussi mentioned by Ian Agol in the comments is a nice one.

I'll finish off by buying into the premise of the question for a moment and mentioning a couple of algebraic applications.

  1. Bridson and Grunewald answered a question of Grothendieck by exhibiting a map between a non-isomorphic pair of residually finite, finitely presented groups, which induced an isomorphism on profinite completions. (As in Francesco's answer, the importance of residual finiteness here was that the \'etale fundamental group is the profinite completion.) They made direct use of Wise's construction of certain residually finite hyperbolic groups.

  2. Let $M$ be a hyperbolic manifold. By the proof of the Virtual Haken theorem, we know that the first Betti numbers of finite covers of $M$ can be taken as large as we like. I believe that number theorists would be very excited if, in the case where $M$ is arithmetic, the same result could be proved using the congruence covers of $M$. Clearly the Virtual Haken theorem, while not enough for them, is a good start!

  3. I hope we can agree that the congruence subgroup problem for quaternionic hyperbolic lattices is of independent algebraic interest. If every hyperbolic group were residually finite then it would follow that the congruence subgroup property fails in this context. (Conversely, proving that they do have the congruence subgroup property might be the best chance to prove that there exist non-residually finite hyperbolic groups.)

...

There are others, but this seems like enough for now.

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  • $\begingroup$ I'll add that, in response to the first question, I think we care less than we used to about whether or not all hyperbolic groups are residually finite. In practice, many of the examples we can build are not just hyperbolic but virtually special, and so have all the 'profinite' properties we might want. $\endgroup$ – HJRW Feb 28 '15 at 8:54
  • $\begingroup$ But this is an example of a question becoming less important because approaches to it are successful, which surely is evidence for the importance of the original question, rather than against it. $\endgroup$ – HJRW Feb 28 '15 at 9:16
  • $\begingroup$ Another important application would be to constructing families of expanders. Since we're good at constructing hyperbolic groups with property T, and Margulis proved that the Cayley graphs of the finite quotients of a property T group form a family of expanders, the residual finiteness of hyperbolic groups would imply the existence of large families of expanders. $\endgroup$ – HJRW Dec 29 '17 at 8:59
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Again in the topological line, Lucks approximation Theorem tells us that if a group is residually finite then we can obtain the L^2 Betti numbers of the group as a limit of the ordinary Betti numbers.

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  1. I addressed your first question in the comments a while ago (we would like to know whether hyperbolic groups have a finite-index torsion-free subgroup, realizing the vcd, which is equivalent to the question of residual finiteness). Another reason that I'm interested in this question is the relation to QCERF, as mentioned by Henry Wilton. I proved that hyperbolic groups which are cubulated are residually finite, and the proof is by an intricate induction on the dimension of the cube complex (in concert with the QCERF property, or rather the stronger local retract property). If hyperbolic groups in general are residually finite, this would obviously subsume my result. Moreover, the proof of such a result would have to be completely different, since there are e.g. hyperbolic groups with property (T), so cannot be cubulated or have the local retract property. So it seems likely that if true, the proof of such a result would entirely subsume my proof in the cubulated case (i.e., it seems unlikely that the proof would depend or build on my proof as a special case). Thus I have spent a bit of time trying to show the existence of non residually-finite hyperbolic groups, to no avail.

  2. I think this is an important question: what does residual finiteness tell us that is not internal to the theory of groups? I don't have an explicit answer to this question. However, certain polynomial diophantine equations can be solved, by first considering solutions on a finite-sheeted covering space, e.g. the primitive solutions to $x^2+y^3=z^7$ were found this way. So in some sense residual-finiteness of the fundamental group comes into play when solving the equation by descent.

Other applications of residual-finiteness (not exactly of the sort that you request, but not mentioned in the other comments or answers):

  • the Kaplansky conjecture holds for group rings of residually finite groups (so "that class of rings have this wonderful property" - ha).
  • iterated monodromy groups (coming from branched covers associated to a rational map) encode information about the dynamics of rational maps, and were used to solve Hubbard's "twisted rabbit problem". These groups are inherently residually finite.
  • Golod and Shafarevich showed that class towers can be infinite by showing that the pro-p completion (quotient?) of a certain Galois group can be infinite. This is related to residual finiteness, although not precisely in the form of your question. Moreover, the question of whether there are infinitely many number fields of class number one would be answered if we knew the existence of infinitely many number fields whose absolute Galois group had a finite maximal solvable quotient.
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The obvious reason is that finite groups are easy to compute with, and infinite groups are not - the fact that the decision problems mentioned are solvable for (some) hyperbolic group is of essentially NO practical interest. For practical purposes, you need residual finiteness, and of the effective kind (see the oeuvre of Khalid Bou-Rabee and yours truly for examples).

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  • $\begingroup$ Do you think it's really true that we need residual finiteness in practice? Or is it just quick and dirty and easy to code? $\endgroup$ – HJRW Mar 2 '15 at 20:58
  • $\begingroup$ @HJRW "need" is a strong word, but I think that without it, it has historically been much more difficult to find practical methods - can't speak for the future, of course. $\endgroup$ – Igor Rivin Mar 3 '15 at 15:23
  • $\begingroup$ I heard a great talk from Colva Roney-Dougal recently (OK, summer 2013, I suppose) about practical algorithms for solving the word problem in negatively curved groups. No doubt she can tell you all about it when you move! $\endgroup$ – HJRW Mar 3 '15 at 20:08
  • $\begingroup$ @HJRW I will be sure to ask her! On a more depressing note, residual finiteness does little for the word problem in f.g. matrix groups. $\endgroup$ – Igor Rivin Mar 3 '15 at 20:29
  • $\begingroup$ Out of interest, Igor, do you ever find yourself (in practice) in the situation where you have a finite presentation of a matrix group but no explicit faithful representation? $\endgroup$ – HJRW Mar 4 '15 at 10:22
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The question has been answered in several different ways. One question of interest is whether a finitely generated group is linear (can be embedded in $GL_n({\mathbb C})$). This implies residual finiteness, by a result of Selberg. So this is one way to test for linearity.

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    $\begingroup$ I thought this result was normally attributed to Mal'cev (I know Mal'cev proved a more general result). $\endgroup$ – user68579 Mar 18 '15 at 20:00
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A common technique of study is to focus on a subset of the properties or a portion of the object under consideration; by understanding a small piece, one may be able to say something about the whole, at least at a local level. So for studying commutativity, it helps to pick two arbitrary elements and look at the (subgroup, except I will switch to universal algebra mode now, and say) subalgebra they generate. Similarly, using congruence conditions modulo various primes, one can say some things about the solutions to various Diophantine equations.

A theorem in general algebra is Garrett Birkhoff's Subdirect Representation theorem: any algebraic structure is a subdirect product of subdirectly irreducible algebras. (See e.g. Algebras, Lattices, Varieties by McKenzie, McNulty, Taylor for definitions and a precise statement.) This gives us a possible tool for study: for the group or ring or cylindric algebra of interest, project it via a homomorphism onto a small subdirectly irreducible component, and see what can be said on that component.

For me, the big picture is what analytical tools I can use. For you, the big picture may be how the large structure is put together. Residual finiteness, like subdirect representation, is a tool that you can use to determine whether and how examining the algebra locally will help.

Gerhard "Remember Picture In Picture Feature" Paseman, 2015.02.26

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  • $\begingroup$ I've received an indication to clarify. When looking at a problem, several perspectives are often useful. The notion of residual finiteness allows one to shift between or adopt several perspectives. This is one aspect of "picture in picture" (a feature on some TV monitors). Another is that this answer attempts to surround the questioner's "big picture" for group theory with an "even bigger picture" of general algebra. I find the signature useful in suggesting a point germane to and indirectly addressing the question. Gerhard "Signatures Are Important To Me" Paseman, 2015.02.26 $\endgroup$ – Gerhard Paseman Feb 26 '15 at 20:25

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