  The intergalactic medium (IGM) is believed to contain the majority of baryons
in the universe and to trace the same dark matter structure as galaxies,
forming filaments and sheets. Lya absorbers, which sample the neutral component
of the IGM, have been extensively studied at low and high redshift, but the
exact relation between Lya absorption, galaxies and the large-scale structure
is observationally not well-constrained. In this study, we aim at
characterising the relation between Lya absorbers and nearby overdense
cosmological structures (galaxy filaments) at recession velocities Delta v \leq
6700 km/s by using archival observational data from various instruments. We
analyse 587 intervening Lya absorbers in the spectra of 302 extragalactic
background sources obtained with the COS installed on the HST. We combine the
absorption-line information with galaxy data of five local galaxy filaments
originally mapped by Courtois et al. (2013). Along the 91 sightlines that pass
close to a filament, we identify 215 (227) Lya absorption systems (components).
Among these, 74 Lya systems are aligned in position and velocity with the
galaxy filaments, indicating that these absorbers and the galaxies trace the
same large-scale structure. The filament-aligned Lya absorbers have a ~90
percent higher rate of incidence (dN/dz=189 for log N HI \geq 13.2) and a
mildly shallower slope (-beta = -1.47) of the column density distribution
function than the general Lya population at z=0, reflecting the filaments'
matter overdensity. The strongest Lya absorbers are preferentially found near
galaxies or close to the axis of a filament, although there is substantial
scatter in this relation. Our sample of absorbers clusters more strongly around
filament axes than a randomly distributed sample would do (as confirmed by a KS
test), but the clustering signal is less pronounced than for the galaxies in
the filaments.
