Coarse group theoretic study on stable mixed commutator length

Abstract

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 $\varphi$.Comment: 69 pages, no figure. Minor revision (v2): some symbols change