We investigate the properties of the H i Ly- \alpha absorption systems ( Ly- \alpha forest ) within and around galaxy voids at z \lesssim 0.1 . We find a significant excess ( > 99 per cent confidence level , c.l . ) of Ly- \alpha systems at the edges of galaxy voids with respect to a random distribution , on \sim 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 ( N _ { HI } ) and Doppler parameter ( b _ { HI } ) distributions of Ly- \alpha systems found inside and at the edge of galaxy voids at the \gtrsim 98 and \gtrsim 90 per cent c.l . respectively . Low density environments ( voids ) have smaller values for both N _ { HI } and b _ { HI } than higher density ones ( edges of voids ) . These trends are theoretically expected and also found in gimic , a state-of-the-art hydrodynamical simulation . Our findings are consistent with a scenario of at least three types of Ly- \alpha systems : ( 1 ) containing embedded galaxies and so directly correlated with galaxies ( referred as ‘ halo-like ’ ) , ( 2 ) correlated with galaxies only because they lie in the same over-dense LSS , and ( 3 ) associated with under-dense LSS with a very low auto-correlation amplitude ( \approx random ) that are not correlated with luminous galaxies . We argue 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 under-dense LSS absorbers account for 25 - 30 \pm 6 per cent of the current Ly- \alpha population ( N _ { HI } \gtrsim 10 ^ { 12.5 } cm ^ { -2 } ) while the other two types account for the remaining 70 - 75 \pm 12 per cent . Assuming that only N _ { HI } \geq 10 ^ { 14 } cm ^ { -2 } systems have embedded galaxies nearby , we have estimated the contribution of the ‘ halo-like ’ Ly- \alpha population to be \approx 12 - 15 \pm 4 per cent and consequently \approx 55 - 60 \pm 13 per cent of the Ly- \alpha systems to be associated with the over-dense LSS .