PSL(2, C), THE EXPONENTIAL AND SOME NEW FREE GROUPS

We prove a normal form result for the groupoid of germs generated by PSL(2, C) and the exponential map. We discuss three consequences of this result: (1) a generalization of a result of Cohen about the group of translations and powers, which gives a positive answer to a problem posed by Higman; (2) as proof that the subgroup of Homeo(R, + infinity) generated by the positive affine maps and the exponential map is isomorphic to an HNN-extension; (3) a finitary version of the immiscibility conjecture of Ecalle-Martinet-Moussu-Ramis.