Suppose is a group. A homogeneous quasimorphism on is a quasimorphism on that is also a 1-homomorphism of groups. Explicitly, a function is termed a homogeneous quasimorphism (sometimes abbreviated as hqm) if and only if it satisfies both these conditions:
- The quasimorphism condition: There exists a positive real number such that for all . Note that depends on but not on the choice of or . The smallest value of that works is termed the defect of and is denoted .
- The 1-homomorphism of groups condition: For any and , we have .
- Given any quasimorphism, we can construct a homogeneous quasimorphism from it called its homogenization. For a quasimorphism , the homogenization is defined as .
- The collection of homogeneous quasimorphisms on a group is a vector space with pointwise addition and scalar multiplication of functions. This vector space is denoted . The collection of all quasimorphisms (including the non-homogeneous ones) is a bigger normed vector space, and is denoted .
- The quotient space is a Banach space with the defect being the norm.