Is it permissible for a husband to prevent his wife from leaving the house by beating her, and does her leaving without his permission and her denial of it make it permissible for him to beat her, and what are the reasons that permit beating one's wife?

A wife leaving her husband's house without his permission, for no necessity, is considered nushuz (rebellion/disobedience). Physical discipline is only to be administered after admonition and abstaining from intimacy in bed, and if these are not effective.

Among the actions considered nushuz that permit physical discipline (with its specific guidelines) are: neglecting adornment for the husband, refusing his call to bed, neglecting prayer and ghusl, leaving the home, and also: striking the husband's female slave, striking a child, insulting or humiliating the husband, insulting a stranger, uncovering the face to a non-mahram, or giving away something from the husband's house without his permission if it is not customary to do so.

Furthermore, refusing to live with the husband and preventing intimacy is also considered nushuz.