1) Ignoring the house for the moment, she can leave her abuser whenever she wants, but cannot cut him 100% out of her life due to the child they have together: she can't keep him seeing his child (if wants to), but would also be entitled to child support if she is the main custodial parent.2) As to the house: whenever non-spouses own a home together and cannot decide mutually what to do with it (e.g. one wants to sell, but the other will not agree to either sell to third parties or to buy her out), a co-owner can bring a kind of legal action in county court commonly called an action "for partition" (that was the term at common law; your state may have a different name for it). In an action for partition, the court can order that the house be sold and the proceeds (after paying the costs of the sale and paying off any mortgages, liens, etc.) be divided between the owners.