On distortion in groups of homeomorphisms

Abstract

Let $X$ be a path-connected topological space admitting a universal cover.Let Homeo$(X, a)$ denote the group of homeomorphisms of $X$ preservinga degree one cohomology class $ a$. We investigate the distortion in Homeo$(X, a)$. Let $g\in$ 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 ahomeomorphism $g$ admits two nonequivalent invariant measures then itis undistorted. We also define a local rotation number of a homeomorphismgeneralizing the notion of the rotation of a homeomorphism of the circle.Then we prove that a homeomorphism is undistorted if its rotation number isnonconstant.