We study the stability of homomorphisms between topological (abelian) groups. Inspired by the "singular" case in the stability of Cauchy's equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω: 𝒢 → ℋ such that ω(0) = 0 and
ω(x+y) - ω(x) - ω(y) → 0
(in ℋ) as x,y → 0 in 𝒢. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x)-a(x) → 0 in ℋ as x → 0 in 𝒢. Our main result is that quasi-homomorphisms ω:𝒢 → ℋ are approximable in the following two cases:
∙ 𝒢 is a product of locally compact abelian groups and ℋ is either ℝ or the circle group 𝕋.
∙ 𝒢 is either ℝ or 𝕋 and ℋ is a Banach space.
This is proved by adapting a classical procedure in the theory of twisted sums of Banach spaces. As an application, we show that every abelian extension of a quasi-Banach space by a Banach space is a topological vector space. This implies that most classical quasi-Banach spaces have only approximable (real-valued) quasi-additive functions.