Commutators of diffeomorphisms, III: a group which is not perfect

Abstract

The group of diffeomorphisms of the real line with compact support is perfect if r~ 2 (cf. [1-3]). It is unknown whether this is the case if r -- 2 (cf. [4]). In this note, we will give a very simple proof that the group G of compactly supported C 1 diffeomorphisms of the real line whose first derivative has bounded variation is not perfect. For f~ G, log Df is a compactly supported function of bounded variation. Let D log Df denote the derivative of log Df in the sense of the theory of distributions. It is well known that D log Df is a compactly supported Radon measure. In other words, if we think of D log Df as a linear functional on the space of C~ functions on , then D log Df has a unique linear continuous extension to the space of continuous functions on R, where we provide this last space with the C ~ topology. A self homeomorphism f of R induces automorphism f* of the continuous functions on R, defined by f*u = u of. The dual of f* is an automorphism f, of the space of compactly supported Radon measures on R. Another way of describing f. is to observe that if X is a Borel subset of R, then (f./~)(X) = ~(f-lX), for any Radon measure tL. If u is a compactly supported function of bounded variation, then