  The presence of magnetic fields in galaxy clusters has been well established
in recent years, and their importance for the understanding of the physical
processes at work in the Intra Cluster Medium has been recognized. Halo and
relic sources have been detected in several tens clusters. A strong correlation
is present between the halo and relic radio power and the X-ray luminosity.
Since cluster X-Ray luminosity and mass are related, the correlation between
the radio power and X-ray luminosity could derive from a physical dependence of
the radio power on the cluster mass, therefore the cluster mass could be a
crucial parameter in the formation of these sources. The goal of this project
is to investigate the existence of non-thermal structures beyond the Mpc scale,
and associated with lower density regions with respect to clusters of galaxies:
galaxy filaments connecting rich clusters. We present a piece of evidence of
diffuse radio emission in intergalactic filaments. Moreover, we present and
discuss the detection of radio emission in galaxy groups and in faint X-Ray
clusters, to analyze non-thermal properties in low density regions with
physical conditions similar to galaxy filaments. We discuss how SKA1
observations will allow the investigation of this topic and the study of the
presence of diffuse radio sources in low density regions. This will be a
fundamental step to understand the origin and properties of cosmological
magnetic fields.
