Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL

Shuhei Maruyama; Takahiro Matsushita; Masato Mimura

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arXiv · 2203.09221

Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL

Shuhei Maruyama·Takahiro Matsushita·Masato Mimura

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Abstract

We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to those of the Poincaré translation number. By using them, we show that the stable commutator length $\mathrm{scl}_G$ and the stable mixed commutator length $\mathrm{scl}_{G,N}$ are not bi-Lipschitzly equivalent for the surface group $G=π_1(Σ_{\ell})$ of genus at least $2$ and its commutator subgroup $N = [π_1(Σ_{\ell}), π_1(Σ_{\ell})]$. We also show the non-equivalence for a pair $(G,N)$ such that $G$ is the fundamental group of a $3$-dimensional closed hyperbolic mapping torus. These pairs serve as the first family of examples of such $(G,N)$ in which $G$ is finitely generated.

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BibTeX RIS

Shuhei Maruyama, Takahiro Matsushita, Masato Mimura. 2023-02-07. Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL. https://arxiv.org/abs/2203.09221

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