Homogeneous quasimorphism

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Suppose G is a group. A homogeneous quasimorphism on G is a quasimorphism on G that is also a 1-homomorphism of groups. Explicitly, a function \mu:G \to \R is termed a homogeneous quasimorphism (sometimes abbreviated as hqm) if and only if it satisfies both these conditions:

  1. The quasimorphism condition: There exists a positive real number D such that |\mu(xy) - \mu(x) - \mu(y)| \le D for all x,y \in G. Note that D depends on \mu but not on the choice of x or y. The smallest value of D that works is termed the defect of \mu and is denoted D(\mu).
  2. The 1-homomorphism of groups condition: For any x \in G and n \in \mathbb{Z}, we have \mu(x^n) = (\mu(x))^n.


  • Given any quasimorphism, we can construct a homogeneous quasimorphism from it called its homogenization. For a quasimorphism f :G \to \R, the homogenization \mu_f is defined as x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}.
  • The collection of homogeneous quasimorphisms on a group G is a vector space with pointwise addition and scalar multiplication of functions. This vector space is denoted Q(G). The collection of all quasimorphisms (including the non-homogeneous ones) is a bigger normed vector space, and is denoted \hat(Q)(G).
  • The quotient space Q(G)/\operatorname{Hom}(G,\R) is a Banach space with the defect being the norm.