My son is not married to the mother of his son. She is cheating and my son wants to take his son and move to another city with full and sole custody.

Your son will first need to establish that he is the biological father of the child. Until that happens, he has no legal rights towards the child whatsover, under Wisconsin law. if the child was born out of wedlock, either he and the mother can enter into a voluntary acknowledgment of paternity and file in Madison, or either of them can file a formal paterntiy action in court. If a formal paterntiy action, either party can request DNA testing to definiteively show whether he is the biological father or not. If he is adjudicated as being the child's father, the court will hold a subsequent hearing, what is sometimes referred to as a "terms hearing," for purpoess of determining custody, placement, visitatio, child support, medical expenses, health insurance and other related finanicial issues affecting the child. You referred to the term "sole custody," but under Wisconsin law, there is a  presumption of joint custody.  "Custody" refers to decision making, not placement, and joint would mean each parent has equal rights on major decisions to be made for the child, such as health, education and religion. If your post means he wants the child to live primarily with him, that is probably going to be very difficult, unless he can show the mother is unfit and incapable of taking care of the child. There is also a law in Wisconsin that prohibits the removal of the child from the state, or more than 150 miltes within the state from the other parent.