Getting Sole Custody in CT?

I was in a relationship with a guy who told me he has been arrested a couple times. Come to find out he has been arrested more than just a couple times. His record JUST in Connecticut consists of Larceny 2 x2, Assault 2, Assault 3, Disorderly Conduct, && Risk Of Injury to a Minor/ Illegal Sexual Contact- Victim. A year later I am just finding this stuff out. And thats something I DO NOT want around my son. So i suppose what i am asking is what are my chances of getting sole custody? Oh hes currently in jail for an assault and larceny & because I helped spend the money (which I THOUGHT was his) I have a Larceny 3 on my record. But thats all I have.