On distortion in groups of homeomorphisms

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving a degree one cohomology class a.
We investigate the distortion in Homeo(X,a). Let g∈ Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invariant Borel probability measures on X and prove that if a homeomorphism g admits two nonequivalent invariant measures then it is undistorted. We also define a local rotation number of a homeomorphism generalizing the notion of the rotation of a homeomorphism of the circle. Then we prove that a homeomorphism is undistorted if its rotation number is nonconstant.