If rent no longer rises then the profit doesn't rise commensurately with demand, so the money isn't there to build supply.

Rent control crushes supply to the degree that the rent control caps rents below the actual market rate.

If you're lucky enough to get rent controlled housing then you're in good shape. But good luck getting it when things get far enough down the timeline.

Rent fixing is the other side where supply is manipulated to extract the maximum profit. In this case the coordinated rent pricing action behaves like an oligopoly.