When money grew on trees during the late great credit boom, private equity firms plunged headlong into New York City real estate. Not only did these companies snag dazzling Manhattan office towers, they also paid up for thousands of mundane rental apartments across the five boroughs. Sure, they had taken on monumental debt to buy these properties, but they had a potent strategy. If they were able to jack up the apartments’ rents, even on those that had much lower, regulated rates, they’d have no trouble profiting mightily. The most famous such bet was the $6.3 billion purchase in 2006 of Stuyvesant Town and Peter Cooper Village on the East River in Manhattan. The buyer was a partnership that Tishman Speyer Properties and BlackRock Realty oversaw. But last week, the properties — now valued at less than $2 billion — went back to the banks that had financed this top-of-the-market deal. The investors in the project had defaulted.