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If he already has the permanent green card (for 10 years, not the conditional 2 year) he doesn't need to be married to apply for citizenship. Unless he does something illegal, the only way to stop him from applying for citizenship or renewing his green card in 10 years would be to report marriage fraud. You will be investigated for possible accessory to fraud as well. Unfortunately you filing for divorce will not help your case.
As long as he is in the country he can be order to pay child support. But if he is deported, nobody can make him.
If you have a proof that he committed fraud you can report him to the immigration office. To get a child support you will need a court order. He will be served a notice. His wages, bank account and federal and state refunds can be garnished to pay of the child support.
There's a office almost in every major state. Click on the state you live in on the bottom of the page.