Re: personal issues and training
I am sorry to hear of this situation. I think your friend is being harsh if she requires that you leave your dojo because of her relationship with this person. How did she know that he is in your dojo?
I think maybe she feels in some way that you are not as sympathetic as you appear to be over this person's behavior. Probably that's a result of low self-esteem from having an abusive boyfriend. Let her know that you are entirely supportive of her and believe his behavior was reprehensible. Although she had problems with him, let her know that you like your dojo and it's not fair for you to have to leave (further continuing his abuse) and you don't have control over who is accepted into the dojo. Unless this person has a criminal history, I don't know that you have a case for having him thrown out.
Is it possible that he joined because he knows you are friends and somehow wants to continue some contact with he, through you?
If he's a reasonable person, you might want to talk to him and let him know of your friendship, if he does not already. Let him know your feelings, and that your friendship is suffering because of this. I know that seems like a risk, but he may decide to leave the dojo on his own. I don't think it's wise to tell other people about his behavior. There is sure to be some tension between you and he and he will pick up on if he does not know you are a friend of his ex-girlfriend.
Otherwise, things might not work out for him at the dojo and he may leave in time on his own.
Best of luck with a difficult situation.