However, both the observer at the center axis and the observer at the perimeter would agree that the circumferance of the perimeter is $2\pi R$.

That is not true for the observer at the perimeter. He/she is moving with an acceleration, and from his/her point of view, space would be distorted, gravitational force, time dilation and other relativistic effects would appear. If he/she would line the perimeter with measuring rods which are motionless from his/her point of view, then he/she would find that it takes more than $2\pi R$ total length of the rods.

Refer to some solid textbook on GR for all those effects. Misner, Thorne, Wheeler is one of most popular.

For the answer by PMay:

However, both the observer at the center axis and the observer at the perimeter would agree that the circumferance of the perimeter is $2\pi R$.

That is not true for the observer at the perimeter. He/she is moving with an acceleration, and from his/her point of view, space would be distorted, gravitational force, time dilation and other relativistic effects would appear. If he/she would line the perimeter with measuring rods which are motionless from his/her point of view, then he/she would find that it takes more than $2\pi R$ total length of the rods.

Refer to some solid textbook on GR for all those effects. Misner, Thorne, Wheeler is one of most popular.