论文标题
Bavard的二元定理用于混合换向器长度
Bavard's duality theorem for mixed commutator length
论文作者
论文摘要
令$ n $为组$ g $的普通亚组。如果$ f（gxg^{ - 1}），$ n $上的准晶体$ f $是$ g $ -invariant，则每个$ g \ in G $中的每个$ g \ and n $ in n $ in g $ in g $ in $ g \ f（x）$。本文的目的是建立Bavard的双重定理，以$ G $ invariant的准态性，以前是川崎和Kimura在此情况下证明的。 我们的二元定理提供了$ g $ invariant的准晶体和$（g，n）$ - 换向器长度之间的连接。在[g，n] $，$（g，n）$ - 换向器长度$ \ mathrm {cl} _ {g，n}（x）$ $ x $的最低数字$ n $，因此$ x $是$ x $是$ n $ commutators的产品，它们是$ n $ commutators，它们是$ n $ commutators，它们是$ n $ n $，在[g，n）$ in [g，n）$ in [g，n）$ in [g，n）$ in $ x $ in [g，n）$ inst in in $ x $ in [g，n）$ in了。在证明中，我们给出了$（g，n）$ - 换向器长度的几何解释。 作为我们的bavard二元性的应用，我们在一对$（g，n）$上获得了足够的条件，根据该条件，$ \ mathrm {scl} _g $和$ \ mathrm {scl} _ {g，n} $是bi-lipschitzly as bi-lipschitzly in $ [g，n] $。
Let $N$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $N$ is $G$-invariant if $f(gxg^{-1}) = f(x)$ for every $g \in G$ and every $x \in N$. The goal in this paper is to establish Bavard's duality theorem of $G$-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura in the case $N = [G,N]$. Our duality theorem provides a connection between $G$-invariant quasimorphisms and $(G,N)$-commutator lengths. Here for $x \in [G,N]$, the $(G,N)$-commutator length $\mathrm{cl}_{G,N}(x)$ of $x$ is the minimum number $n$ such that $x$ is a product of $n$ commutators which are written as $[g,x]$ with $g \in G$ and $h \in N$. In the proof, we give a geometric interpretation of $(G,N)$-commutator lengths. As an application of our Bavard duality, we obtain a sufficient condition on a pair $(G,N)$ under which $\mathrm{scl}_G$ and $\mathrm{scl}_{G,N}$ are bi-Lipschitzly equivalent on $[G,N]$.
