This is family law question and not a family violence question. There are certainly some factors in your favor.
Family Violence Lawyer
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Maybe. The overriding factor will be what is in the best interest of the child. If the child is in an unstable environment, abandoned and not being cared for properly, than you have good cause to seek custody of the child. You need to hire an attorney to discuss this further. I handled these types of cases a lot in Travis County and I also offer a free initial consultation.