Borel Ranks and Wadge Degrees of Context Free Omega Languages

We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter B"uchi automata have the same accepting power than Turing machines equipped with a B"uchi acceptance condition. In particular, for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free languages accepted by 1-counter B"uchi automata, and the supremum of the set of Borel ranks of context free omega languages is the ordinal gamma^1_2 which is strictly greater than the first non recursive ordinal. This very surprising result gives answers to questions of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In:“A Decade of Concurrency”, LNCS 803, Springer, 1994, p. 583-621].