Amenable equivalence relations and Turing degrees

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Abstract
In [12] Slaman and Steel posed the following problem:Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order preservingly in <d for a cone of d's?They had already obtained a partial answer to this question by showing that there is no such d ↦ <d with <d of order type ζ = ω* + ω on a cone. Already the possibility that <d has order type ζ · ζ was left open.We use here, ideas and methods associated with the concept of amenability (of groups, actions, equivalence relations, etc.) to prove some general results from which one can obtain a positive answer to the above problem.

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