Reconstructing quasimorphisms from associated partial orders and a question of Polterovich

TL;DR

This paper demonstrates that continuous homogeneous quasimorphisms on simple Lie groups can be reconstructed from associated partial orders, linking invariant orders with bounded cohomology and enabling concrete computations.

Contribution

It establishes a method to reconstruct quasimorphisms from partial orders, extending to infinite-dimensional Lie groups and connecting to bounded cohomology.

Findings

Reconstruction of quasimorphisms via partial orders

Concrete computation of relative growth in Lie groups

Link between invariant orders and bounded cohomology

Abstract

We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry.