I'm a Peruvian citizen with a US permanent residence card (green card). I entered Mexico by car, using my expired passport. Can I travel back to the US by airplane with the passport which expired a year ago? I came to Mexico because my wife is Mexican and has US citizenship.
You do not need a passport to travel back to the US, but you will need a valid US Permanent Resident Card (Green Card) with you. If you've been out of the US for more than a year, you'll also need a Reentry Permit or Returning Resident Visa and may want to consult an immigration lawyer for assistance in making your case that you haven't abandoned permanent residency.
However, to fly to the US, you also need to satisfy the airline's requirements for identification. You'll need to confirm that they will allow you to check-in using only a permanent residency card. Otherwise, you'll have to contact the Peruvian Embassy or Consulate in Mexico to get your passport renewed. You could also cross the land border using only your permanent resident card.