The Divergence were a race that would have risen to power after the Time Lords. Rassilon didn't want to commit genocide, so he locked them away in a time loop to be trapped until the end of time. "Divergence" was a name given to the creatures by Rassilon. It is unclear what their actual name was, if they had any at all.
Biology
Rassilon did not know whether the Divergence were a race, a species, or an entire genus. (AUDIO: Zagreus)
History
The activities of the Divergence before their exile by Rassilon are unknown. However, after their exile to the Divergent Universe, they promptly set themselves up as its dominant species and set to work trying to break the cycle of death and rebirth. This included setting up a base of operations on the planet Caerdroia, and constructing the "crucible world" of Bortresoye to conduct their experiments. (AUDIO: The Next Life)
The Divergence got their revenge on Rassilon for destroying their future when he also was banished into the Divergent Universe. There, the Divergence subjected him to the torments of the cycle which he had created at least eighty-four times.
Eventually the cycle somehow caused the Divergence to wither and die but they were then absorbed by the creature known as Daqar Keep, who subsequently absorbed all their power and assumed their authority in the Divergent Universe. (AUDIO: The Next Life)
Attempts to escape
In Cardington, 1951, Doctor Stone and Reverend Townsend opened a doorway to the Divergent Universe during a military experiment. The Divergence/Keep attempted to escape, but the portal failed before they reached it, killing Stone and Townsend.
Near the end of time, approximately sixty billion years in the future, "Uncle Winky's Wonderland" (an amusement park) had been built over Rassilon's Foundry, which held the deteriorating main portal to the Divergent Universe. Again the Divergence/Keep attempted to escape, but the portal was closed by Uncle Winky before they could get through. (AUDIO: Zagreus)