They have Turkmenistan.
They are muslim invaders in Greek Anatolia.
“In the early Middle ages, Turkmens called themselves Oghuz and in the Middle Ages they took the ethnonym Turkmen.[18] These early Oghuz Turkmens moved westward from the Altai Mountains through the Siberian steppes, and settled in the region now known as Turkmenistan. Further westward migration of the Turkmen tribes from the territory of modern Turkmenistan and the rest of Central Asia started from the 11th century and continued until the 18th century. These Turkmen tribes played a significant role in the ethnic formation of such peoples as Anatolian Turks”
The Turks came from parts East, in several waves: first the Seljuk Empire, then the various emirates that ended up being incorporated into the Ottoman Empire.There were two stages in the Turkification of Anatolia: 1071 (Battle of Manzikert) through ca. 1300, and 1300 through ca. 1500.
In the first stage, there was conquest, massacres, migration, economic disruption, and forced conversions. Yet in 1300, there were still enough Christians in the emirates in Anatolia, that the capital tax on Christians was their largest source of revenue.
The population of Christians in Western Anatolia plummeted over the next two centuries, so that by the early 16th century the proportion of Christian to Muslim households in the Anatolia Eyalet was 8000 to 520,000. (In the Rûm Eyalet by contrast, which had remained under Christian control up to 1461, Christian households were still 30%.)
The population of Christians plummeted, of course, because Greek-speaking Christians became Turkish-speaking Muslims. Greek clergy continually refer to their flock dwindling, and to Christians converting. The number of bishoprics in Anatolia dwindled from 54 to 17, with depopulated sees merged into their neighbours.
Not only was the Greek church much less empowered to retain the allegiance of Christians, with its property confiscated and its prestige diminished, but there was significant Islamic missionary activity as well (the Bektashi Order, the Mevlevi Order, the Futuwwa).