We investigate the properties of the H i Lyα absorption systems (Lyα forest) within and around galaxy voids at z ≲ 0.1. We find a significant excess (>99 per cent confidence level, c.l.) of Lyα systems at the edges of galaxy voids with respect to a random distribution, on ∼5 h−1 Mpc scales. We find no significant difference in the number of systems inside voids with respect to the random expectation. We report differences between both column density () and Doppler parameter () distributions of Lyα systems found inside and at the edge of galaxy voids at the ≳98 and ≳90 per cent c.l., respectively. Low-density environments (voids) have smaller values for both and than higher density ones (edges of voids). These trends are theoretically expected and also found in Galaxies-Intergalactic Medium Interaction Calculation (GIMIC), a state-of-the-art hydrodynamical simulation. Our findings are consistent with a scenario of at least three types of Lyα systems: (1) containing embedded galaxies and so directly correlated with galaxies (referred to as ‘halo-like’), (2) correlated with galaxies only because they lie in the same overdense large-scale structure (LSS) and (3) associated with underdense LSS with a very low autocorrelation amplitude (≈random) that are not correlated with luminous galaxies. We argue that the latter arise in structures still growing linearly from the primordial density fluctuations inside galaxy voids that have not formed galaxies because of their low densities. We estimate that these underdense LSS absorbers account for 25–30 ± 6 per cent of the current Lyα population ( cm−2), while the other two types account for the remaining 70–75 ± 12 per cent. Assuming that only cm−2 systems have embedded galaxies nearby, we have estimated the contribution of the ‘halo-like’ Lyα population to be ≈12–15 ± 4 per cent and consequently ≈55–60 ± 13 per cent of the Lyα systems to be associated with the overdense LSS.