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There is an intimate relation between entanglement entropy and Riemann surfaces. This fact is explicitly noticed for the case of quadratic fermionic Hamiltonians with finite range couplings. After recollecting this fact, we make a comprehensive analysis of the action of the M\"obius transformations on the Riemann surface. We are then able to uncover the origin of some symmetries and dualities of the entanglement entropy already noticed recently in the literature. These results give furtherdoi:10.1088/1742-5468/2016/04/043106 fatcat:2qwsvy3znbd2xmb37gxxtf7tce