Edge

According to the documentary featured on the second disc of the Advent Children set,
a new urban township has grown around the devastation of Midgar. This town is called "Edge."
Director Tetsuya Nomura admits that this information was omitted from the film.

This omission strikes me as hilarious, especially in light of the facts
that (1) Advent Children is hailed as art and (2) the directorial control
of narrative flow (which includes exposition of relevant information, such as
setting and backstory) are crucial for any elementary attempt at storytelling.

At any rate, the name "Edge" indeed appears on a number of
signs through the course of the film.

This is taken from a street sign in the urban area.

This is taken from the background during the scene in which Tifa 
discovers Denzel around the central monument, while 
the Seph-Spawn Kids summon monsters.