Is it permissible to buy an apartment for the mother with the husband's money without his knowledge, claiming it is a gift from the mother, with the husband later becoming aware that it is his property, due to the father's ill-treatment and the mother's fear of divorce?

Regarding your father's oath of divorce: If your mother is certain that he has exceeded three divorces, it is not permissible for her to remain with him. She must separate from him, even if through khul' or by referring the matter to the judiciary. If she is not certain, she must refer her case to scholars.

Regarding your father preventing you from taking anything from his house: There is no sin upon him for that, because your sustenance is obligatory upon your husband, not your father. As for your father taking from your money: If he is poor and you are rich, his sustenance is obligatory upon you, according to what is customary. If he is rich, it is not permissible for him to take anything except with your willing consent, and it is commendable for you to be good to him. If you are poor, nothing is obligatory upon you. It is not permissible for you to take from your husband's money to give to him except with his permission.

As for taking from your husband's money without his knowledge to buy an apartment for your mother, this is impermissible and unlawful, because it is disposing of his money without his permission. It is not permissible for you to spend from his money except with his knowledge, except for what is customary to be overlooked. What you intend to do, writing the apartment in his name as a gift from your mother, does not justify the act and it remains unlawful.