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?