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The police officers who patrol New York City’s transit system are constantly scanning subway stations looking for a familiar set of faces: repeat sexual offenders who have arrest records.

The officers follow them through jam-packed platforms and onto crowded trains, waiting to see if they grope, grind, molest or indecently expose themselves again.

Because only then can the police intervene and make an arrest or eject them from the subway system.

The authorities have long been frustrated by not being able to keep known subway sexual harassers from returning to the scene of their crimes.

But now, lawmakers and police officials are pushing a dramatic measure: On Monday, a city councilman, Chaim M. Deutsch, introduced a bill that would bar repeat sexual offenders from the subway system for life.