Homogenization of a quasimorphism
From Groupprops
Definition
Suppose is a quasimorphism. The homogenization of is a homogeneous quasimorphism (sometimes denoted ) defined as follows:
Facts
Defect
- Upper bound: The defect of the homogenization is at most twice the defect of .
- Lower bound: There is no effective lower bound besides the obvious one of zero. To see this, note that any quasimorphism with bounded image has homogenization the zero map, even though we can have quasimorphisms with bounded image that have arbitrarily large defect.