Reconstructing Quasimorphisms from Associated Partial Orders and a Question of Polterovich

We show by elementary combinatorial arguments that any non-zero homogeneous quasimorphism on a given group can be realized as the relative growth of some bi-invariant partial order on that group. Thereby we provide a link between quasimorphisms, bounded cohomology and partial orders. This yields existence results for bi-invariant partial orders, e.g. for non-elementary Gromov hyperbolic groups. Given continuous quasimorphism of topological groups, one would like to realize them as relative growths of continuous orderings. It turns out that this problem is surprisingly different from its discrete counterpart. We provide a positive solution in the case of finite-dimensional simple Lie groups, based on Lie semigroup theory. As an application we show that the metric space associated by Eliashberg and Polterovich to the minimal bi-invariant continuous partial order on a simple Lie group collapses. We extend our method to prove a similar collapsing result for certain orders on H̃am(CP).