Consider asynchronous induction motor as precise servo axis

The short answer is, "yes", with a few caveats:

This sort of performance is available only in conjunction with a high-performance flux-vector variable frequency drive controller. Three variable frequency drive controllers up to the task are:
a.) Gozuk EDS AC flux-vector variable frequency drive controller.
b.) ABB ACS-800 AC drive controller with direct-torque control.
c.) Emerson - Control Techniques "Uni-Drive" in full flux-vector mode.

You need solid closed-loop feedback to get high-performance, position-compensated control, but you also have to carefully-design the application. Some things to consider are:

d.) Response. Do you need this motor to, "stand on its ear", and, "stop on a dime"?
e.) Inertia: If possible, you need to calculate the mass moment of inertia: especially, taking into account the reflected inertia.
f.) Overall feed rate: Solid rules of thumb are to design the axis to operate at no less than half the rated, base motor velocity, and to design maximum gear-in velocity to occur at 95-% of the command reference. This is critical: you have to allow the servo axis to be able to correct at the maximum feed rate.

g.) Design tip: Never, under any circumstance, treat the drive controller as a, "variable-modulo gear reducer". Too many companies over-advertise their variable frequency drive controllers, especially full-blown servo amplifiers as having the capability of operating at ultra-low velocity gradients. They probably can, and initially, the solution will probably work well, because the mechanics are stiff. Lose that stiffness, and now, you're in a pickle.

In closing, unless you're looking for resolutions in the neighborhood of 0.05-RPM, you don't have to employ a full-blown servo amplifier / motor combination.