Coarse geometry of stable mixed commutator length I: duality and functional analysis on chains

TL;DR

This paper explores the duality and functional analysis of stable mixed commutator length in groups, characterizing when it aligns with ordinary stable commutator length via invariant quasimorphisms.

Contribution

It introduces a refined duality theorem and analyzes the functional structure on chains to understand the geometry of stable mixed commutator length.

Findings

Characterizes bi-Lipschitz equivalence of stable commutator lengths via invariant quasimorphisms.

Develops a refined version of the generalized mixed Bavard duality theorem.

Performs functional analysis on the completion of a space of 1-chains.

Abstract

Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length $\mathrm{scl}_{G,N}$ and the restriction of the ordinary stable commutator length $\mathrm{scl}_{G}$ are defined. We characterize when they are bi-Lipschitz equivalent by the vanishing of a certain $\mathbb{R}$-linear space $\mathrm{W}(G,N)$ related to invariant quasimorphisms. For the proof, we obtain a refined version of the generalized mixed Bavard duality theorem, and perform functional analysis on the completion of a certain space of $1$-chains.