Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. More precisely, $f$ is a quasimorphism if there exists some constant $C$ such that for every $x,y \in \Gamma$, $$|f(xy)-f(x)-f(y)|<C$$ In other words, $f(xy)$ is at bounded distance from $f(x)+f(y)$.

A natural question is whether any quasimorphism is simply a homomorphism perturbed by a bounded function. In this regard, we can define the bounded cohomology of $\Gamma$ with coefficients in $\mathbb{R}$, and consider the comparison homomorphism $$c:H_b^2(\Gamma,\mathbb{R}) \to H^2(\Gamma,\mathbb{R})$$ and it is known that the kernel of this map is precisely the space of non-trivial quasimorphisms. More precisely $$ker(c) = QM(\Gamma)/(Hom(\Gamma,\mathbb{R}) \oplus C_b(\Gamma,\mathbb{R}))$$ where $QM(\Gamma)$ is the space of quasimorphisms, $C_b(\Gamma,\mathbb{R})$ is the space of all bounded functions, and $Hom(\Gamma,\mathbb{R})$ is the space of homomorphisms.

So when $ker(c)$ is trivial, every quasimorphism is trivial: it is obtained by perturbing a homomorphism by a bounded function.

Is there a quantitative version of this statement? That is, suppose $C$ is the uniform bound for the quasimorphism $f$ as above, and suppose $ker(c)$ is trivial. Then can we actually give a concrete bound for the distance of $f$ from the homomorphism in $Hom(\Gamma,\mathbb{R})$ in terms of $C$?

All I find in references is that $f$ can be written as a sum of a homomorphism and a bounded function, but I have not come across any actual bound on that function in terms of the original bound on $f(xy)-f(x)-f(y)$. This seems like a very natural question, so do help out with directions and references if available.

Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. More precisely, $f$ is a quasimorphism if there exists some constant $C$ such that for every $x,y \in \Gamma$, $$|f(xy)-f(x)-f(y)|<C$$ In other words, $f(xy)$ is at bounded distance from $f(x)+f(y)$.

A natural question is whether any quasimorphism is simply a homomorphism perturbed by a bounded function. In this regard, we can define the bounded cohomology of $\Gamma$ with coefficients in $\mathbb{R}$, and consider the comparison homomorphism $$c:H_b^2(\Gamma,\mathbb{R}) \to H^2(\Gamma,\mathbb{R})$$ and it is known that the kernel of this map is precisely the space of non-trivial quasimorphisms. More precisely $$ker(c) = QM(\Gamma)/(Hom(\Gamma,\mathbb{R}) \oplus C_b(\Gamma,\mathbb{R}))$$ where $QM(\Gamma)$ is the space of quasimorphisms, $C_b(\Gamma,\mathbb{R})$ is the space of all bounded functions, and $Hom(\Gamma,\mathbb{R})$ is the space of homomorphisms.

So when $ker(c)$ is trivial, every quasimorphism is trivial: it is obtained by perturbing a homomorphism by a bounded function.

Is there a quantitative version of this statement? That is, suppose $C$ is the uniform bound for the quasimorphism $f$ as above, and suppose $ker(c)$ is trivial. Then can we actually give a concrete bound for the distance of $f$ from the homomorphism in $Hom(\Gamma,\mathbb{R})$ in terms of $C$?

All I find in references is that $f$ can be written as a sum of a homomorphism and a bounded function, but I have not come across any actual bound on that function in terms of the original bound on $f(xy)-f(x)-f(y)$. This seems like a very natural question, so do help out with directions and references if available.

*UPDATE: So in one direction it seems to be easy. Suppose $f=\phi+g$ where $\phi \in Hom(\Gamma,\mathbb{R})$ and $g:\Gamma \to \mathbb{R}$ is a bounded function with $|g(x)|<D$ for every $x \in \Gamma$. Then $$|g(xy)-g(x)-g(y)|<3D$$ by the triangle inequality, and $$|f(xy)-f(x)-f(y)|\leq |\phi(xy)-\phi(x)-\phi(y)|+|g(xy)-g(x)-g(y)|$$ and so $$|f(xy)-f(x)-f(y)|< 3D$$ So if $f$ is globally $D$-close to a homomorphism, then it is locally $3D$-close to being a homomorphism, or a $3D$-quasimorphism. But I am interested more in the other direction.