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A tunable slow light with wide bandwidth, large group index, and low dispersion is realized in polymer-infiltrated slot photonic crystal waveguide (PI-SPCW). The slow-light properties of PI-SPCW are first optimized by shifting the first and second rows of air holes adjacent to the slot. Using the three-dimensional (3-D) plane wave expansion method, the low dispersion slow light with nearly constant group indices of 55, 100, 172.5, and 222 over bandwidths of 9.0, 3.5, 1.1, and 0.8 nm are demonstrated. Then, the dynamic modulation of the optimized slow light in PI-SPCW is investigated. Thanks to the electro-optic effect of the infiltrated polymer, the central wavelength of flatband slow light could be flexibly tuned by the external driving voltage. For the slow light with high group index of 100, the modulation sensitivity up to -0.75 nm/V is observed and the effective bandwidth could be further enlarged to 11 nm. These achievements open the possibility for achieving a wide range of targeted optical functionalities in PI-SPCW that can be dynamically controlled according to the practical requirements and highlight the versatility, flexibility, and tunability offered by electro-optic effect in PI-SPCW.