Custody Laws in New York?

Hi there, I'm needing some help from maybe some people who have been there before. I am a mother to an also three year old child. The father has not been in her life since she was five weeks old and took off. It's always been a worry of mine that he would one day file custody papers to see his child even though that has never happen. There's a past history of abuse against me which I still have the police reports for. My question is what can I file against him to terminate his rights as a father? Is there any laws in new york state that protect parents like me ? Could i file child abandonment? I've been with my now fiance for two years and he would love to adopt her so were would i began?