The G-Bi-shadowing property with nilpotent group

In this paper, we investigated the property of bi-shadowing for actions of certain finitely generated groups. We have given a definition of the term Bi-Conditions and mentioning some key definitions with example. We have formulated the text of the reductive bi-shadowing theory (through which we set sufficient conditions for the action to have the bi-shadowing property) and we try to prove it by formulating and proving some lemmas that complement one another in order to arrive at proving the main theorem in the paper.