ERBIL, IRAQ — Syria’s Kurds have mostly escaped prolonged bouts of direct conflict in the country’s civil war, but with rebel units pushing east toward the resource-rich Kurdish heartland, Kurdish militias proliferating and calls for greater autonomy growing, this may not remain the case.

Last summer, the Democratic Union Party, known by its Kurdish-language acronym P.Y.D., seized control of many towns and villages in the Kurdish majority northeast. The group also holds territory in a few Aleppo neighborhoods and some towns around the city.

The P.Y.D. is the most powerful Kurdish faction in Syria and has a well trained militia. This is perhaps a product of its ties to the Kurdistan Workers’ Party, or P.K.K., a guerrilla group that has been fighting for Kurdish autonomy in Turkey.

The leadership of the P.Y.D. plays down its ties to the P.K.K. But Syrian Kurds often use the names interchangeably, and P.Y.D. offices feature portraits of the imprisoned P.K.K. leader Abdullah Ocalan and Syrian P.K.K. guerrillas killed in fighting with Turkey.