Does a divorce occur with the husband's statement, "You are not under my responsibility if you do such-and-such," if the wife denies having done it, intending during her marital life, while he intends the action in absolute terms, and is she obligated to inform him of the sin after repenting from it?

A husband's questioning his wife about her past and demanding that she admit to falling into forbidden acts is impermissible. She must conceal her past and not inform her husband of her sin. If he asks her about it, she is permitted to lie to him, because lying is permissible for the purpose of concealing a sin. However, since the husband's statement is an ambiguous phrase that could imply divorce, she must inform him that she lied to him and ask him about his intention behind that phrase. If he intended divorce, then a divorce has occurred.