I thought the major problem with SF isn't lack of rent control but the inability to build up. Don't they have a lot of laws preventing larger/taller apartment complexes? Get more renters into an area and the rent should control and stabilize itself better.

I'm not too informed about this so please correct me if I'm wrong.

That is part of the problem. But largely it's just because it's a desirable place to live. Manhattan has plenty of residential skyscrapers and that hasn't made rent go down.