I have a friend who is in a very verbally abusive relationship.
Her husband ( they are not legally married) is constantly out never goes anywhere with his family plays video games all the time when he IS home. Travels a lot doesn't take the family with him and when they do try to communicate it ends up in a fight. If she asks how his day was at work and says I am your wife I have a right to communication, he yells and says I don't have to answer your questions. ( My friend is still not a legal resident of the US) The last argument they had over why they don't communicate ended with him cursing the day they met which really hurt her feelings.
She threatened to leave him he said "you can leave all you want the baby stays with me, you will never see our 2 year old daughter I promise you that" Can you please let me know what her rights are as a mother ( technically she is considered a single mom, they never got legally married) But she is not a legal resident.

Ooops... she is not a legal resident but her husband whom she has never been married... :confused:

If she asks how his day was at work and says I am your wife I have a right to communication, he yells and says I don't have to answer your questions.
Correct. He is not under obligation to do it.
She threatened to leave him he said "you can leave all you want the baby stays with me, you will never see our 2 year old daughter I promise you that" Can you please let me know what her rights are as a mother
Having said she is an illegal resident it may / or may not / have impact on a judge. No one can predict what will happen.

Her paperwork is at the immigration and in process she doesn't have a green card yet but has authorization to work in the US ( if that makes a difference)

First, he is not her husband an she is not his wife. She needs to understand that. Second, while she may not be a legal resident of the US, she is a documented alien. Which may afford her some rights.

You say she has authorization to work in the US, but does she have a job?

Is the father (not husband) on the birth certificate?

There is no question in my mind that she needs to leave. The only question is whether she should wait until her immigration status is finalized. I think she should. Once that is done, she should leave and file immediately for custody and child support. In the meantime she should document his threats about not letting him see the child. This may help limit his chances of visitation.

She should also try and contact some local support groups for abused women (though I don't really see her as abused, just in a lifeless relationship) to get help with the custody stuff.

Well, here is the actual part of it they had a wedding they just weren't registered because they constantly put it off She has a part time job, and is taking care of her child constantly he will not let her put their dausghter in Daycare saying that she shouldn't work and if she does then his mom is a better caregiver ( she MIL leaves diapers on for hours and the baby gets a rash) My friend has no say in the family and every time she talks he says you got married for making voice? Watch your kid otherwise you'll never see her again.

Either they were married by a civil authority or they weren't. If they were then she has to file for divorce.