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It follows that for any z 1 in the upper half plane, there is an element g 1 in the subgroup Γ 1 of Γ generated by T such that w 1 = g 1 (z 1) satisfies λ ≤ Re w 1 ≤ μ, i.e. this strip is a fundamental domain for the translation group Γ 1. If |w 1 | ≥ 1, then w 1 lies in X and the result is proved.