TITLE: Residual finiteness: why do we care?
QUESTION [26 upvotes]: 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.

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.)
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?

REPLY [5 votes]: 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. 
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.