I'd venture a guess and say I'm very much an outlier in the HN crowd, because I consider myself a socialist. Even with that said, I don't really understand the long-term logic behind rent caps, and I don't think it does much for fighting the old trope that "socialists are economically illiterate" either.
If you have a finite supply of something, like housing, it seems pretty obvious that the reason the price is going up is because the supply is unable to catch up with demand. Not only do more people want to live in the city, they're also willing and able to pay more. The mechanics are obviously more complex because this is all affected by things like interest rates, influx of foreign capital and whatever, but I think the basic assumption still stands: there's not enough housing.
So if we approach this problem from a "neo liberal" and a "socialist" point of view, I think there are two ways you'd normally go about it, none of which include capping rents because that achieves nothing except make people hoard the commodity that was previously rare, and is now even more rare because of market manipulation:
Neo-lib: The market will take care of it, and poor people being pushed out is not a problem, they can live somewhere cheaper and new housing will be built because builders want to make money.
Socialist: Poor people have a right to live here too, but there is a lack of affordable housing. We should tax the rich to enable a wider array of housing financed by the state.
Regardless of which extreme (or somewhere in between you are) I still can't understand what rent caps would achieve apart from being a destructive policy for almost everyone except the ones who already are lucky enough to have a contract. You don't get more houses to live in by making the houses a more valuable commodity through market manipulation.