Tire with radial carcass reinforcement having a tread of thickness E, this tread having a tread surface and at least one cut opening onto the tread surface to form edge corners. The cut extends into the tread over a total depth H. The cut is in the form of a sipe having a width D. There is also formed, in the vicinity, i.e. at a minimum distance from the said edge corner that is at most equal to five times the width D of the cut, of at least one edge corner of this cut and on the tread surface, at least one cavity having a depth h that is small in comparison with the depth of the cut, i.e. that is at most equal to 30% of the depth H. The at least one cavity reduces the compression rigidity of the tread in the vicinity of the the at least one edge corner.