How can you get a synthetic population to grow.

Use the Assembly projects. They build your population.

There is a small chance that I am mistaken, but a synthetic race really needs the Yor tech tree, otherwise you don't have access to the assembly projects and can't increase your population

To the OP:

The only way you can increase the population of colonies whose population has capped as a synthetic faction using a non-Yor tech tree is to take advantage of the fact that docking a loaded colony or transport ship with a colonized planet adds the population carried by the colony or transport ship to the population of the colony (you can scrap the ships while they're docked if you want to get rid of them, or launch the ships so as to reuse them; if you launch the ships, be aware that there's a minimum population taken when launching colony and transport ships). This will allow you to overpopulate planets, and if I recall correctly you'll only lose 0.1 population per turn at overpopulated colonies.

All synthetic factions should have the basic assembly project, regardless of what tech tree they use. What the Yor tech tree does for synthetic factions that no other tech tree in the unmodded game does is provide access to better assembly projects and to ways to increase the synthetic population cap.