Is expiation (fasting three days or feeding ten needy persons) obligatory before the husband can engage in intercourse with his wife, given that he pronounced her divorce three times with the intention of threat, and that the wife is refraining from marital relations because of him?

It is advisable to avoid swearing by divorce, as it is an oath of the wicked. You have made your wife's divorce contingent three times. If the contingent condition does not occur, then nothing is incumbent upon you. If it does occur, then the divorce becomes binding according to the number of instances in which the condition was violated. The occurrence of divorce upon the fulfillment of the contingent condition is the view of the majority of scholars, and expiation for an oath (kaffarat yamin) is not sufficient for it. However, according to Shaykh al-Islam Ibn Taymiyyah, expiation for an oath is required, which is feeding ten poor persons, clothing them, or freeing a believing slave. If one is unable, then fasting for three days. The fatwa adopted by us is the view of the majority. There is no relation between giving expiation and enjoying one's wife. It is not permissible for the wife to refuse intimacy without a legitimate excuse. It is advisable to appear before a Sharia judge or someone whose knowledge is trusted to clarify the details of the contingent divorce.