On proper extensions of the conformal group of sphere diffeomorphisms

Abstract

In this paper, we prove that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of M\"obius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. In addition, we show that any such group must always contain an element of positive topological entropy. We also provide an elementary characterization, in terms of transitivity, of the M\"obius transformations within the full group of sphere diffeomorphisms.