What is required to keep the peace (for now) with a fallen empire?

There are four types of Fallen Empire, and each one has different things that anger them, based on their ethics.

Keepers of Knowledge (Fanatic Materialists): The Fallen Empire dedicates itself to the hoarding of technologies that it believes should be kept out of the hands of young and irresponsible races. Pursuing dangerous lines of research will draw their ire. (Technologies considered "dangerous" are marked in red)

Notably, AI and Jump Drives

Enigmatic Observers (Fanatic Xenophiles): Empires that use slaves, or purge populations may suffer the wrath of the Observers.

Holy Guardians (Fanatic Spiritualists): This fallen empire dedicates itself to the preservation and defense of its holy sites. Colonizing systems that they consider sacred is likely to incur their ire.

Militant Isolationists (Fanatic Xenophobe): The fallen empire dedicates itself to the defense of its borders against any intrusions. Colonizing systems that border them is likely to draw their ire.

Other things, such as insulting, rivalling them or declaring war (obviously) will anger them as well. Importantly, they will not declare war on you until relations drop below -75

Source: Stellaris Wiki

In my current game I was surrounded by two Militant Isolationists empires. I could expand right next to their borders without any problems but only by keeping up an embassy in both realms. Kept their happiness at around -30 which is way above the required -75.