AXES-SDSS: comparison of SDSS galaxy groups with all-sky X-ray extended sources

We revisit the picture of X-ray emission of groups through the study of systematic differences in the optical properties of groups with and without X-ray emission and study the effect of large-scale density field on scaling relations. We present the identification of X-ray galaxy groups using a combination of RASS and SDSS data. We include new X-ray reanalysis of RASS, to include very extended (up to a size of half a degree) sources and account for differences in the limiting sensitivity towards compact and very extended X-ray emission. X-ray groups exhibit less scatter in the scaling relations and selecting the groups based on the extended X-ray emission leads to an additional scatter reduction. Most of the scatter for the optical groups is associated with a small (6%) fraction of outliers, primarily associated with low optical luminosity groups found in dense regions of the cosmic web. These groups are the primary candidates for being the contaminants in the optical group catalogues. Removing those groups from the optical group sample using optically measured properties only, leads to a substantial reduction in the scatter in the most scaling relations of the optical groups. We find a density dependence of both the X-ray and optical luminosity of groups, which we associate with the assembly bias. Abridged.