Tire with radial carcass reinforcement having a tread of thickness E, this tread having a tread surface to come into contact with a roadway and comprising at least one cut opening onto the tread surface to form edge corners, this cut extending into the tread over a total depth H at most equal to the thickness E of the tread, this cut being, between the tread surface and a depth H1 at most equal to the total depth H, in the form of a sipe having a width D suitable for being able to close up at least in part under normal running conditions of the tire. 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, this at least one cavity reducing the compression rigidity of the tread in the vicinity of the said at least one edge corner.