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Question:
Were Jews cut off from the Babylonian Geonim?
The Islamic Schism was a secondary reason for the waning influence of the Babylonian Geonim. The primary reason was economic, and is tied to the decline and troubles incurred by the city of Baghdad itself.
History of Baghdad
Baghdad's early meteoric growth slowed due to troubles within the Caliphate, including relocation of the capital to Samarra (during 808–819 and 836–892), the loss of the western and easternmost provinces, and periods of political domination by the Iranian Buwayhids (945–1055) and Seljuk Turks (1055–1135).
As Bagdhad's fortunes waned so did the the independence of the Geonim, which became entirely dependent upon donation from abroad.
Gaon
The decline of the Baghdad caliphate, the impoverishment of Babylonian Jewry which caused the academies to depend completely on contributions from abroad, the greatness and the independent intellectual development of the Diaspora, and the persecutions by the Abbasid and Seljuk rulers put an end to the institution of the gaonate in about 1040.