Does he even deserve to see her?

My daughter is 6 yrs old. I met her dad when i was very young. And naive. And stupid. I married him after she was born when I turned 18. After we got married, he became physically abusive (before we got married, it was emotional and mental abuse). We separated when my daughter was 2. He never laid a hand on her, only ever spanked her once. She of course doesn't remember how he was. After I moved out he had a few gfs that I did not get along with, (one did drugs, the other picked fights with me and called me a bad mother, and the last one had 2 kids she didnt have custody of so i absolutely did not trust her). I always gave him the option to see our daughter anytime he wanted as long as he didnt bring his gfs to the visits. He refused to come without them. So the visits got far and few between. I even made attempts to take her to a neutral location without me there and he always had an excuse to cancel. He got to where everytime she'd try to call he wouldnt answer and hardly ever even called her back. Well, I met my new husband and had a son with him. My daughter calls my new husband "daddy". Well lately, because my ex isnt in a relationship, he's tried to make a little more contact (not much really). Hes seen her at Christmas and her birthday a week ago and only 1 phone call between that where WE called HIM. He kept feeding me his b.s. lines about wanting to try to be a part of her life but he STILL makes no effort. He refuses to sign over his rights even though I have promised and promised that he could still see her anytime he asked if he did. He of course blames me why he doesn't get to see our daughter. I'm just tired of all of it. I hide all of thise from our daughter so when she does see him, its like he's the only person in the room. I don't blame her but I get so envious and pissed. I mean, is it truly worth trying anymore if he isn't even going to?

( Sorry for typing a novel for you to read, if you made it this far, I would really love some advise?)