We present the results obtained with the test bench of the Geometry Monitoring System (GMS) for the ALICE Muon Spectrometer. It consists in a mock up, reproducing at full scale, three half planes of the chambers 6, 7 and 8 of the spectrometer. We show that the GMS is able to measure transverse displacements with an accuracy of 1.5 microm. We show also that the resolution deteriorates by a factor 3 to 4 when thermal gradients are generated