To own a region, your faction must conquer the Fortress at the center of the region and hold it through Contested Period to win the ownership of the Region.
Once the region is owned by your faction, its color on Mini-map will change to indicate its owning faction.
IMPORTANT: If your city suffers a massive defeat in the region owned by an enemy faction, it will be teleported away to a random location! This is not limited to the area of Lava Cracks!
And if your city suffers a massive defeat on Lava Cracks (regardless the ownership of the region), it will always be teleported away to a random location.