From my experience the Arcology just took time to increase, I don't remember doing anything in particular to get it from Level 1 to Level 8. The Arcology just grows itself. I know this isn't a concrete answer and there could be something happening behind the scenes, but that's my experience. I wish there would have been some kind of challenge to actually get to maxed out, but there isn't.
I raised my population of the Arcology, even while staying at level 8. I didn't start any new city, didn't really raise my population anywhere else, it just grew by itself. I have come to the realization that this game does a horrible job of communicating what goes on behind the scenes.