The April 2007 eruption of Piton de la Fournaise was the biggest volcano eruptive crisis of the 20th and 21st centuries. Interferometric synthetic aperture radar (InSAR) captured a large coeruptive seaward displacement on the volcano's eastern flank, which continued for more than a year at a decreasing rate. Coeruptive uplift and posteruptive subsidence were also observed. While it is generally agreed that flank displacement is induced by fault slip, we suggest that this flank displacement might have been induced by a sheared sill, based on observations of sheared sills at Piton des Neiges. To test this hypothesis, we develop a new method to invert a quadrangular curved source submitted to simultaneous pressure and shear stress changes. This method, based on boundary elements, is applied to data acquired along six Envisat orbits covering a 14 month period subsequent to the April 2007 eruption. Posteruptive displacement is well explained by closure and slip of a large (5 km by 8 km) and shallow (500 m) trapezoidal fracture parallel to the flank and probably coincident with a lithological discontinuity. We investigate whether thermal contraction or degassing of a coeruptive sill can explain the displacement. Such a sill would have to be 10 times thicker than inferred from the coeruptive uplift and solidification time 10 times shorter (~20 days) than the duration of the posteruptive subsidence (24 to 33 months). Instead, we propose that the posteruptive eastern flank displacement is due to the compaction and ongoing slow slip on a shallow detachment fault.