KAYSERI, Turkey—In hindsight, it was a soccer match that kindled Mehmet’s hatred of Chinese rule and set him on the path to exile.

In 2002, Mehmet was at university in Xinjiang, the northwest corner of China that is home to his Uighur ethnic group and the source of a wave of deadly violence in the past two years. He and some other Uighurs decided to support Turkey in the soccer World Cup, he said.

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