High rates of evasion have been reported among high-school students, particularly in the city’s suburbs, Mr. Tengblad said. The group has also sprouted sister organizations in Goteborg and Oslo, and is credited with influencing the free-fare protesters of Movimento Passe Livre in Brazil, who carried out demonstrations last year.

Efficiency has improved with time. Planka’s organizers said that 30 to 40 percent of membership dues were typically required to pay off fines. That means that of the 500 paying members, only about 15 reported being caught per month. In 2004, the group said, about 80 percent of fees were needed to cover fines.

There have been public pleas from Swedish authorities and international searches for a dodger-proof turnstile, instructional videos teaching new group members how to evade payment, and at least one fare-beating dog trained to open the barricades for its owner.

That maneuver: Let the animal squeeze through the small space beneath the gates, then coax it to jump up and down until the sensor is set off to open them. Well-developed canine leg muscles are required.

Most other strategies appear to involve closely following a fellow traveler, using a scarf or jacket to set off sensors on the far side of the barriers, or, for the city’s slender evaders, simply slipping through the sliding gates.

The dodgers’ ubiquity seems to have weakened the resolve of those charged with stopping them. On a recent afternoon at Stockholm Central Station, a station agent sipped an energy drink beside the gates as riders filed past.

“If someone does it, like, 20 yards from here,” he said, “I’m not going to run.”

One group member, Alex Berthelsen, 29, said that many guards and fare controllers — employees in yellow suits who are expected to check tickets — had come to recognize him, but still failed to stop him.