An Ergodic Action of the Outer Automorphism Group of a Free Group

Theorem. Suppose that G is a connected group locally isomorphic to a product of copies of SU(2) and U(1). If n > 2, then the Out(Fn)-action on Hom(Fn, G)/G is ergodic. We conjecture that Out(Fn) is ergodic on each connected component of Hom(Fn, G)/G for every compact Lie group G and n > 2. When G = U(1), then this action is just the action of GL(n,Z) on the n-torus R/Z, which is well known to be ergodic. In fact, certain cyclic subgroups of GL(n,Z) act ergodicly. The proof relies heavily on [2], both in its outline and a key result. When n = 2, the action is not ergodic, since it preserves the function