  Let $G$ be a group and $N$ a normal subgroup of $G$. We study the large scale
behavior, not the exact values themselves, of the stable mixed commutator
length $scl_{G,N}$ on the mixed commutator subgroup $[G,N]$; when $N=G$,
$scl_{G,N}$ equals the stable commutator length $scl_G$ on the commutator
subgroup $[G,G]$. For this purpose, we regard $scl_{G,N}$ not only as a
function from $[G,N]$ to $\mathbb{R}_{\geq 0}$, but as a bi-invariant metric
function $d^+_{scl_{G,N}}$ from $[G,N]\times [G,N]$ to $\mathbb{R}_{\geq 0}$.
Our main focus is coarse group theoretic structures of
$([G,N],d^+_{scl_{G,N}})$. Our preliminary result (the absolute version)
connects, via the Bavard duality, $([G,N],d^+_{scl_{G,N}})$ and the quotient
vector space of the space of $G$-invariant quasimorphisms on $N$ over one of
such homomorphisms. In particular, we prove that the dimension of this vector
space equals the asymptotic dimension of $([G,N],d^+_{scl_{G,N}})$.
  Our main result is the comparative version: we connect the coarse kernel,
formulated by Leitner and Vigolo, of the coarse homomorphism $\iota_{G,N}\colon
([G,N],d^+_{scl_{G,N}})\to ([G,N],d^+_{scl_{G}})$; $y\mapsto y$, and a certain
quotient vector space $W(G,N)$ of the space of invariant quasimorphisms. Assume
that $N=[G,G]$ and that $W(G,N)$ is finite dimensional with dimension $\ell$.
Then we prove that the coarse kernel of $\iota_{G,N}$ is isomorphic to
$\mathbb{Z}^{\ell}$ as a coarse group. In contrast to the absolute version, the
space $W(G,N)$ is finite dimensional in many cases, including all $(G,N)$ with
finitely generated $G$ and nilpotent $G/N$. As an application of our result,
given a group homomorphism $\varphi\colon G\to H$ between finitely generated
groups, we define an $\mathbb{R}$-linear map `inside' the groups, which is dual
to the naturally defined $\mathbb{R}$-linear map from $W(H,[H,H])$ to
$W(G,[G,G])$ induced by $\varphi$.

Group Theory;Geometric Topology;Metric Geometry