PARIS – Mary Fallot looks as unlike a terrorist suspect as one could possibly imagine: a petite and demure white Frenchwoman chatting with friends on a cell-phone, indistinguishable from any other young woman in the café where she sits sipping coffee.

And that is exactly why European antiterrorist authorities have their eyes on thousands like her across the continent.

Ms. Fallot is a recent convert to Islam. In the eyes of the police, that makes her potentially dangerous.

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