In geocentric models, did the Earth rotate? In geocentric models, did the Earth rotate around its axis?

I assume that if the Earth rotated around its axis, then the Sun would not have to orbit around the Earth at a very fast rate, whereas if the Earth didn't rotate, then the Sun would have to orbit the Earth every day.
In the historic geocentric models, and in particular in the Ptolemaic model, the earth is immobile in the centre of the cosmos, and the sphere of the fixed stars rotates around it once a day, carrying the sun, moon and planets with it. There were astronomers who realised that this apparent motion could also be explained, from a purely mathematical point of view, by assuming that the earth rotates on its axis, but they rejected this option. By the way, neither Ptolemy nor any other serious astronomer believed that the earth was flat.