Distortion in Groups of Circle and Surface Diffeomorphisms

In his seminal article [18] S. Smale outlined a program for the investigation of the properties of generic smooth dynamical systems. He proposed as definition of the object of study the smooth action of a non-compact Lie group G on a manifold M ; i.e., a smooth function f : G ×M →M satisfying f(g1, f(g2, x)) = f(g1g2, x) and f(e, x) = x for all x ∈ M and all g1, g2 ∈ G, where e is the identity of G. Equivalently one can consider the homomorphism φ : G → Diff(M) from G to the group of diffeomorphisms of M given by φ(g)(x) = f(g, x). The primary motivation, and by far the most studied case, has been that where G is either the Lie group R of real numbers or the discrete group Z. As noted in the Introduction to this volume this study grew out of an interest in solution of differential equations where the group R or Z represents time (continuous or discrete). In this article we will focus on the far less investigated case where G is a subgroup of Lie group of dimension greater than one. The continuous and discrete cases when G is R or Z share many characteristics with each other and