You may be able to pursue a dissolution of marriage and should do so if your are truly unhappy in your marriage. However, you would need to consult a matrimonial/family law attorney for the process in the jurisdiction where you live. Each state has its own requirements and processes for seeking a divorce.
You are not required to sign the I-751 regardless of your marital status. However, your spouse may seek a waiver of the joint petition requirement. A divorce in and of itself does not mean the alien would not be able to obtain his/her permanent green card.
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A petition for dissolution of marriage is simply a document that you file with the Court to start a divorce case. It must be in the proper form and state details of your residency and what relief you are seeking. After you file the petition with the Court, you must have your spouse presented with a copy, and they have the right to file a Response form, contesting your request. They cannot stop you from getting divorced from them, but they can argue over the term - thing like division of property, and custody of your children, if you have any together. You can read more about these laws here: http://www.northwestlawoffice.com/divorce.html
As I'm sure you're aware, divorce can have immigration implications. But if your spouse is being abusive to you, then I would advise you not to worry about their immigration status. If you've agreed to sponsor them as an immigrant, you may want to consult with an immigration attorney as well, to be sure you won't be further liable. My office has attorneys who practice in both areas, if you want to contact us.
There are various legal measures that can be taken to ensure your safety during the divorce. Without knowing more details of just what has happened, it's difficult to recommend any one tactic over another. For that reason, you should consult with an attorney in private.
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Do what you need to do to ensure your safety, first and foremost. It sounds as though you're getting started on the divorce paperwork. And, yes, the divorce will impact his immigration status, but he'll still be eligible to get his green card.
This general advice does not create an attorney-client relationship.