Yes you can be outside the United States and retain your permanent residence status however there are limitation and if you are out for more than 6 months at one time you may be considered to have abandoned your status. You may formally relinquish your status in writing. Immigration has a form for this.
The answer provided here is general in nature and does not take into account other factors that may need to be reviewed for a more precise answer. You should consult with an immigration attorney before taking any action. The answer here is not intended to create an attorney-client relationship.
Being a permanent resident of the U.S. you are expected to spend most of your time inside the U.S. If you don’t you run the risk of having your permanent residence status terminated, and if your Green Card is cancelled by the immigration officer, it will be difficult to obtain permanent residence status again.
For this reason, some LPRs who stay outside the U.S. for long periods of time, give up their Green Card voluntarily, usually seeking to do so before it can be cancelled. The one clear advantage of giving up your Green Card voluntarily is you can apply again in future if you want to.
However, keep mind that if you relinquish your lawful permanent resident status but wish to regain it in the future you must qualify all over again for the status and abandonment of lawful permanent resident status is irrevocable.
Also, each time you want to come to the U.S., and you need a new visa, you will have to go to the consulate for your visa interview.
Surrendering your green card should be done through the U.S. Consulate General or Embassy in your nation of residence. Once you surrender your green card, it cannot be revived.
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You should apply for a reentry permit as it is the best evidence that you did not intend to abandon your status when you left the U.S. They are normally good for two years.
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