If you are applying for a green card on the basis of your marriage to your boyfriend, then living apart can affect your application because the burden is on you and your husband to prove to that were married in good faith. Some of the factors that USCIS looks at when making a decision is whether you are actually living together and sharing your lives. Having TPS will not speed up the process.
Answered on Feb 20th, 2014 at 4:44 AM