ISRAEL JOURNAL OF MATHEMATICS, Vol. 30, Nos. 1-2, 1978 ON TEICHMOLLER'S THEOREM ON THE QUASI-INVARIANCE OF CROSS RATIOS* BY IRWIN KRA ABSTRACT Teichmiiller's theorem gives necessary and sufficient conditions for mapping one ordered quadruple by a K-quasiconformal map onto a second ordered quadruple. We give a simple non-computational proof of the necessity part. We then characterize such extremal mappings, and obtain as a consequence a new formula for the modular function, which leads to a very simple derivation of the known expression for the Poincar6 metric on the thrice-punctured sphere. a* a* w Let (a~,a2, a3, a,) and (a*, 2, 3, a*) be two ordered quadruples of distinct points in the extended complex plane, C U {~}. Form the cross ratios -- m 3 a = a3-a~__ . a3 a4 a * = __a* a* a*-a*. a2-al a2-a4' a*-a* " a*-a* (Of course, ct is the image of al under the M6bius transformation that sends a3, a4, a2 to 0, 1, o0.) The cross ratios are points in the twice-punctured plane C\{0, 1}. Let r ) denote the non-Euclidean Poincar4 distance on C\{0,1}. It is obtained by projecting the Poincar4 metric on the unit disc A 1 1-1zl21&r to C \ {0, 1}. If the corresponding distance is denoted by p ( .,. ), then by choosing a holomorphic universal covering map ~r: a--*C\{O, 1}, we have * Research partially supported by NSF grant MCS 76-04969A01. Received August 4, 1977 152