I find it interesting that everyone sees this as a sliding scale with culture on one side and rent on the other. There's a third variable that the city could play with to keep the rent down while not building a single building:

Raise Taxes

If you want to lower prices and you can't/won't increase supply, the obvious solution is to lower demand. Adding a sharp jump in the local income tax at a high income level will free up housing space as wealthy citizens leave the city. The less affluent won't see a tax increase, since it's outside their bracket, but they will see housing become available, lowering rent prices.

Of course, this will be a hit on the economy as all the wealthy individuals leave the area, but it decreases rent while keeping culture constant.