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In a class of Type-II seesaw dominated $SO(10)$ models proposed recently withheavy neutrinos, extra $Z'$ boson, and resonant leptogenesis, at first we showthat the lightest first generation sterile neutrino that mediates dominantcontributions to neutrinoless double beta decay also generates the displacedvertex leading to verifiable like-sign di-electron as well as di-muonproduction events outside the LHC detectors having suppressed standard modelback-ground and missing energy. Resonant leptogenesis in this case isimplemented by a pair of quasi-degenerate sterile neutrinos of the second andthe third generations having masses of ${\cal O}(500)$ GeV. Then we predict anew alternative scenario where the models allow the second generation sterileneutrino mass to be ${\cal O}(10)$ GeV capable of mediating the dominant doublebeta decay as well as the displaced vertices for significantly improved numberof like-sign dilepton events in different channels. Resonant leptogenesis inthis alternative scenario is mediated by a pair of heavy quasi-degeneratesterile neutrino masses of the first and the third generations. In addition toQD type light neutrino mass hierarchy, we also show how these results arederived for the NH case. We also discuss $Z'$ production cross sections at$\sqrt s =14 $ TeV run-II of LHC and also at ILC. While the lepton flavourviolating branching ratios are only few to four orders less than the currentexperimental bounds, proton lifetime predictions are accessible to ongoingSuper K. or Hyper K. searches.

Dilepton events with displaced vertices, double beta decay, and resonant  leptogenesis with Type-II seesaw dominance, TeV scale $Z'$ and heavy  neutrinos