论文标题
在圆圈上作用于圆的组的不变的准晶体和SCL的非等效性
Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL
论文作者
论文摘要
我们为在圆圈上作用的群体构建不变的准畸形。此外，我们为所产生的准畸形的非扩展性和明确的公式提供了标准，该公式将我们的准态值与庞加莱翻译数字的值相关联。通过使用它们，我们表明稳定的换向器长度$ \ mathrm {scl} _g $和稳定的混合换向器长度$ \ mathrm {scl} _ {g，n} $不是表面组$ g =π_1（n poss $ g =π_1（n poss）$ g =π_1$ ns $ ns $ n n $ bi-lipschitzly等于[π_1（σ_{\ ell}），π_1（σ_{\ ell}）] $。我们还显示了一对$（g，n）$的非等效性，因此$ g $是$ 3 $二维的封闭双曲线映射圆环的基本组。这些对是这样的$（g，n）$的第一个家族，其中$ g $有限地生成。
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.
