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.
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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.)
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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?
Best Answer
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.