In this paper, we investigate the absorption and dispersion properties of a weak probe field in a triple quantum well nanostructure by using the incoherent pumping fields. A deep 7.1 nm-thick GaAs well is coupled, on one side, to two shallow 6.8 nm-thick Al0.2Ga0.8As wells by a 2.5 nm-thick Al0.4Ga0.6As barrier. The two shallow wells are separated by a 2.0 nm-thick Al0.4Ga0.8As barrier. Both sides of quantum well contact with 36 nm Al0.4Ga0.6As. Therefore, this type of triple quantum well nanostructure can be used as a suitable medium for studying the effect of spontaneously generated coherence (SGC) and interference between incoherent pumping fields on absorption and dispersion properties of weak probe light. We find that the interferences from spontaneous emission and incoherent pumping processes can change the slope of dispersion and group velocity of the probe light from slow to fast or vice versa. Moreover, it is demonstrated that the group velocity of the light pulse can be controlled with the rates of incoherent pumping fields.