"their hegemony" is correct and usable in written English.
You could use it to describe a situation in which one group has authority, power, or control over another group, such as, "The country's ruling regime maintained their hegemony through oppressive measures."
They see threats to their hegemony on all sides.
The ethnic Uzbeks, Tajiks and Hazaras of the north and west resent their hegemony.
They find it easier to legitimise and extend their hegemony when no other discourses are heard.
This will force the ethnic nation controlling the state to resort to further undemocratic means to maintain their hegemony.
As for absent friends, Rafael Nadal and Roger Federer will be emphatically reminded that their hegemony is over.
How fast they settled on she who is most likely to preserve their hegemony for the foreseeable and unforeseeable future.
The focus was the Nation-State and how the most powerful countries used International Law to maintain their hegemonies.
Avoiding the term "metamodernism," while publicly prizing metamodern literature, is an easy way for late postmodernists to avoid acknowledging that their day--or, at least, their hegemony--has ended.
America doesn't care if New Zealanders buy their flatscreen TVs from China, they care that we watch their shows, their culture, their values and their economic hegemony.
Through an adroit mixture of diplomacy, political maneuvering and canny horse sense, Britain's Lloyd George and France's Clemenceau managed to retain their grip on their colonial empires and their global hegemony for at least another generation.
Verizon is beholden to manufacturers to continue their Droid hegemony (and, most recently, their access to the iPhone) and T-Mobile is no more, swallowed up by the only GSM carrier in the US to sell the iPhone.
