While the custody case is going on, the judge will usually give a temporary custody order that gives the child equal access to both parents as much as is practical unless:
the judge believes that the abuser has a history of committing domestic violence against you, your child, any other child, or a romantic partner the abuser lives with (“domestic living partner”). If the judge believes this, the abuser may not get custody unless s/he meets certain requirements. See Can a parent who committed violence get custody?1