混合换位子长度的Bavard对偶定理
Bavard's duality theorem for mixed commutator length
摘要：
设$N$是群$G$的正规子群。$N$上的拟同构$f$是$G$不变的，如果对每个$x \in N$和每个$g \in G$都有$f(gxg^{-1}) = f(x)$.本文的目的是建立$G$ -不变拟同构的Bavard对偶定理，这是Kawasaki和Kimura在$N = [G,N]$的情况下证明的。我们的对偶定理提供了$G$ -不变拟同构和$(G,N)$ -换位子长度之间的联系.这里对于$x \in [G,N]$，$x$的$(G,N)$ -换向器长度$\mathrm{cl}_{G,N}(x)$是最小数$n$，使得$x$是被写为$[g,x]$的$n$换向器与$g \in G$和$h \in N$的乘积。在证明中，我们给出了$(G,N)$ -换位子长度的几何解释。作为Bavard对偶的应用，我们得到了$(G,N)$对上$\mathrm{scl}_G$与$\mathrm{scl}_{G,N}$在$[G,N]$上双Lipschitzly等价的一个充分条件.
英文摘要：
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 $[G,N]$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
