A couple of interesting replies.
Of course, as President, one must have all the facts. The United States does have an extradition treaty with Turkey, as follows:
http://photos.state.gov/libraries/tu...df/32t3111.pdf
I think Article 3 (Conditions of Refusal) is perhaps applicable.
Of course, as one poster noted by implication, there are wheels within wheels. The President of Turkey just met with Putin of Russia. Relationships between the two powers had been strained, if you will recall, by Turkey having shot down a Russian military jet that Turkey claimed had strayed over its territory (from Syria). Russia imposed sanctions against Turkey. However, things improved after President Erdogan wrote a letter to the family of the dead pilot last month, and President Putin was among the first to express support for Erdogan during the failed coup. The two countries do have close economic ties.
Will not sending back Gulen drive Turkey closer to Russia, including militarily?
Yet, it seems very doubtful that Turkey has any basis for us to extradite Mr. Gulen. I have not seen where he has been charged with breaking any Turkish laws, although I suspect that President Erdogen will take care of that soon. Yet, in accordance with Article 3, above, would we not suspect that any such criminal charge is politically motivated?
As for those that believe that we should simply send Mr. Gulen 'back', from the country that he has lived in since 1999, would that not violate the express treaty? Plus, Mr. Erdogan and Turkey is, rather obviously, headed for a dictatorship. As we have learned long ago, you cannot appease a dictator. Return Mr. Gulen, and Turkey will start demanding more of its exiled citizens living in the United States.
As President, I would stick to the clear language of the Treaty, citing Article 3. Of course, we would inform Turkey through the Turkish embassy here in the United States, or perhaps through our embassy in Ankara.
It seems unlikely, at present, that Turkey would withdraw from NATO (it has been a member since 1952). However, if Turkey and Russia enter into a new alliance, of course Turkey may conclude that 'defense' against Russia is no longer necessary (the purpose, after all, of NATO).