He said to his wife: “If you leave the house, then you are divorced.” Then she left the house, then she came back and told him in an arrogant and sarcastic way that she was divorced
Question

I have been married for three years and now I have a son as well. I and my wife have always had arguments leading up to big fights. She has also left the house a number of times. Once the argument worsened so, that I warned her if she left the house she was divorced. After having said that she left the house and came back and said to me in an arrogant and sarcastic way that she was divorced. Is she really divorced or not? If she is, does she have the right to ask for the MAHR which I have to pay her?
Praise be to Allah.If the man says to his wife, “If you leave the house, then you are divorced,” and his intention is to divorce her, then if she goes out she is divorced from him. 
But if his aim in saying that was the same as swearing an oath, and it was only to rebuke, forbid and threaten, and it was not his intention that divorce should take place, then this is a matter concerning which the scholars differed. The majority are of the view that the divorce does count as such, but some of them are of the view that the divorce does not count in this case, but the one who swore that oath must offer expiation for breaking the oath. This is the correct view that is followed on this website. We have previously stated that in the fatwa referred to above. 
See also the answer to question no. 82400 
Based on that, if your aim when you said this was simply to prevent her from leaving the house, and you did not intend that divorce should take place, then in this case all you have to do is offer expiation for breaking the oath, because the oath was broken when she went out of the house. The expiation for breaking the oath has been discussed in fatwa no. 45676
But if your intention was that divorce should indeed take place if she went out, then in this case the divorce occurred when she went out. If this was the first or second divorce (talaaq), then it is revocable, and it is permissible for you to take her back so long as her ‘iddah has not yet ended, even if she does not want that, because her approval is not needed with regard to taking her back. 
But if this was the third divorce, then she is irrevocably divorced from you and she is not permissible for you until after she has married a husband other than you in a genuine marriage, not one intended to make it permissible for her to go back to you, then he divorces her or dies. 
In either case, whether the divorce was revocable or irrevocable, what you must do is give her her mahr in full, unless she herself waives it or part of it of her own accord. Allah, may He be exalted, says (interpretation of the meaning):
“And give to the women (whom you marry) their Mahr (obligatory bridal money given by the husband to his wife at the time of marriage) with a good heart, but if they, of their own good pleasure, remit any part of it to you, take it, and enjoy it without fear of any harm (as Allah has made it lawful)” [an-Nisa’ 4:4]. 
And Allah knows best.